Calculus Examples

$\tan (x - y) = \frac{y}{8 + x^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\tan (x - y)) = \frac{d}{d x} (\frac{y}{8 + x^{2}})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x - y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x - y$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [x - y]$

Step 2.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [x - y]$

Step 2.1.3

Replace all occurrences of $u$ with $x - y$ .

$\sec^{2} (x - y) \frac{d}{d x} [x - y]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x - y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- y]$ .

$\sec^{2} (x - y) (\frac{d}{d x} [x] + \frac{d}{d x} [- y])$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\sec^{2} (x - y) (1 + \frac{d}{d x} [- y])$

Step 2.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$\sec^{2} (x - y) (1 - \frac{d}{d x} [y])$

Step 2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\sec^{2} (x - y) (1 - y')$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = y$ and $g (x) = 8 + x^{2}$ .

$\frac{(8 + x^{2}) \frac{d}{d x} [y] - y \frac{d}{d x} [8 + x^{2}]}{{(8 + x^{2})}^{2}}$

Step 3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{(8 + x^{2}) y' - y \frac{d}{d x} [8 + x^{2}]}{{(8 + x^{2})}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $8 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [x^{2}]$ .

$\frac{(8 + x^{2}) y' - y (\frac{d}{d x} [8] + \frac{d}{d x} [x^{2}])}{{(8 + x^{2})}^{2}}$

Step 3.4

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{(8 + x^{2}) y' - y (0 + \frac{d}{d x} [x^{2}])}{{(8 + x^{2})}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

$\frac{(8 + x^{2}) y' - y \frac{d}{d x} [x^{2}]}{{(8 + x^{2})}^{2}}$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{(8 + x^{2}) y' - y (2 x)}{{(8 + x^{2})}^{2}}$

Step 3.7

Multiply $2$ by $- 1$ .

$\frac{(8 + x^{2}) y' - 2 y x}{{(8 + x^{2})}^{2}}$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$\frac{8 y' + x^{2} y' - 2 y x}{{(8 + x^{2})}^{2}}$

Step 3.8.2

Reorder terms.

$\frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$\sec^{2} (x - y) (1 - y') = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}}$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by ${(x^{2} + 8)}^{2}$ .

$\sec^{2} (x - y) (1 - y') {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\sec^{2} (x - y) (1 - y') {(x^{2} + 8)}^{2}$ .

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Step 5.2.1.1.1

Apply the distributive property.

$(\sec^{2} (x - y) \cdot 1 + \sec^{2} (x - y) (- y')) {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2.1.1.2

Simplify the expression.

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Step 5.2.1.1.2.1

Multiply $\sec^{2} (x - y)$ by $1$ .

$(\sec^{2} (x - y) + \sec^{2} (x - y) (- y')) {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2.1.1.2.2

Rewrite using the commutative property of multiplication.

$(\sec^{2} (x - y) - \sec^{2} (x - y) y') {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2.1.1.3

Apply the distributive property.

$\sec^{2} (x - y) {(x^{2} + 8)}^{2} - \sec^{2} (x - y) y' {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2.1.1.4

Simplify the expression.

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Step 5.2.1.1.4.1

Reorder factors in $\sec^{2} (x - y) {(x^{2} + 8)}^{2} - \sec^{2} (x - y) y' {(x^{2} + 8)}^{2}$ .

$\sec^{2} (x - y) {(x^{2} + 8)}^{2} - y' \sec^{2} (x - y) {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2.1.1.4.2

Reorder $\sec^{2} (x - y) {(x^{2} + 8)}^{2}$ and $- y' \sec^{2} (x - y) {(x^{2} + 8)}^{2}$ .

$- y' \sec^{2} (x - y) {(x^{2} + 8)}^{2} + \sec^{2} (x - y) {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$ .

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Step 5.2.2.1.1

Cancel the common factor of ${(x^{2} + 8)}^{2}$ .

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Step 5.2.2.1.1.1

Cancel the common factor.

$- y' \sec^{2} (x - y) {(x^{2} + 8)}^{2} + \sec^{2} (x - y) {(x^{2} + 8)}^{2} = \frac{x^{2} y' - 2 x y + 8 y'}{{(x^{2} + 8)}^{2}} {(x^{2} + 8)}^{2}$

Step 5.2.2.1.1.2

Rewrite the expression.

$- y' \sec^{2} (x - y) {(x^{2} + 8)}^{2} + \sec^{2} (x - y) {(x^{2} + 8)}^{2} = x^{2} y' - 2 x y + 8 y'$

Step 5.2.2.1.2

Reorder $x^{2}$ and $y'$ .

$- y' \sec^{2} (x - y) {(x^{2} + 8)}^{2} + \sec^{2} (x - y) {(x^{2} + 8)}^{2} = y' x^{2} - 2 x y + 8 y'$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) {(x^{2} + 8)}^{2} + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2

Simplify $- y' \sec^{2} (x - y) {(x^{2} + 8)}^{2} + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$ .

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Rewrite ${(x^{2} + 8)}^{2}$ as $(x^{2} + 8) (x^{2} + 8)$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) ((x^{2} + 8) (x^{2} + 8)) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.2

Expand $(x^{2} + 8) (x^{2} + 8)$ using the FOIL Method.

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Step 5.3.2.1.2.1

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{2} (x^{2} + 8) + 8 (x^{2} + 8)) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.2.2

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 8 + 8 (x^{2} + 8)) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.2.3

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 8 + 8 x^{2} + 8 \cdot 8) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.3

Simplify and combine like terms.

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Step 5.3.2.1.3.1

Simplify each term.

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Step 5.3.2.1.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.3.2.1.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{2 + 2} + x^{2} \cdot 8 + 8 x^{2} + 8 \cdot 8) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.3.1.1.2

Add $2$ and $2$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{4} + x^{2} \cdot 8 + 8 x^{2} + 8 \cdot 8) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.3.1.2

Move $8$ to the left of $x^{2}$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{4} + 8 \cdot x^{2} + 8 x^{2} + 8 \cdot 8) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.3.1.3

Multiply $8$ by $8$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{4} + 8 x^{2} + 8 x^{2} + 64) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.3.2

Add $8 x^{2}$ and $8 x^{2}$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) (x^{4} + 16 x^{2} + 64) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.4

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - y' \sec^{2} (x - y) (16 x^{2}) - y' \sec^{2} (x - y) \cdot 64 + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.5

Simplify.

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Step 5.3.2.1.5.1

Multiply $16$ by $- 1$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - y' \sec^{2} (x - y) \cdot 64 + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.5.2

Multiply $64$ by $- 1$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) {(x^{2} + 8)}^{2}$

Step 5.3.2.1.6

Rewrite ${(x^{2} + 8)}^{2}$ as $(x^{2} + 8) (x^{2} + 8)$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) ((x^{2} + 8) (x^{2} + 8))$

Step 5.3.2.1.7

Expand $(x^{2} + 8) (x^{2} + 8)$ using the FOIL Method.

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Step 5.3.2.1.7.1

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{2} (x^{2} + 8) + 8 (x^{2} + 8))$

Step 5.3.2.1.7.2

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 8 + 8 (x^{2} + 8))$

Step 5.3.2.1.7.3

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{2} x^{2} + x^{2} \cdot 8 + 8 x^{2} + 8 \cdot 8)$

Step 5.3.2.1.8

Simplify and combine like terms.

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Step 5.3.2.1.8.1

Simplify each term.

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Step 5.3.2.1.8.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.3.2.1.8.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{2 + 2} + x^{2} \cdot 8 + 8 x^{2} + 8 \cdot 8)$

Step 5.3.2.1.8.1.1.2

Add $2$ and $2$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{4} + x^{2} \cdot 8 + 8 x^{2} + 8 \cdot 8)$

Step 5.3.2.1.8.1.2

Move $8$ to the left of $x^{2}$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{4} + 8 \cdot x^{2} + 8 x^{2} + 8 \cdot 8)$

Step 5.3.2.1.8.1.3

Multiply $8$ by $8$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{4} + 8 x^{2} + 8 x^{2} + 64)$

Step 5.3.2.1.8.2

Add $8 x^{2}$ and $8 x^{2}$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) (x^{4} + 16 x^{2} + 64)$

Step 5.3.2.1.9

Apply the distributive property.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) x^{4} + \sec^{2} (x - y) (16 x^{2}) + \sec^{2} (x - y) \cdot 64$

Step 5.3.2.1.10

Simplify.

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Step 5.3.2.1.10.1

Rewrite using the commutative property of multiplication.

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) x^{4} + 16 \sec^{2} (x - y) x^{2} + \sec^{2} (x - y) \cdot 64$

Step 5.3.2.1.10.2

Move $64$ to the left of $\sec^{2} (x - y)$ .

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) x^{4} + 16 \sec^{2} (x - y) x^{2} + 64 \cdot \sec^{2} (x - y)$

$y' x^{2} - 2 x y + 8 y' = - y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) x^{4} + 16 \sec^{2} (x - y) x^{2} + 64 \sec^{2} (x - y)$

Step 5.3.2.2

Reorder factors in $- y' \sec^{2} (x - y) x^{4} - 16 y' \sec^{2} (x - y) x^{2} - 64 y' \sec^{2} (x - y) + \sec^{2} (x - y) x^{4} + 16 \sec^{2} (x - y) x^{2} + 64 \sec^{2} (x - y)$ .

$y' x^{2} - 2 x y + 8 y' = - y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y)$

Step 5.3.3

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) = y' x^{2} - 2 x y + 8 y'$

Step 5.3.4

Move all terms containing $y'$ to the left side of the equation.

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Step 5.3.4.1

Subtract $y' x^{2}$ from both sides of the equation.

$- y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) - y' x^{2} = - 2 x y + 8 y'$

Step 5.3.4.2

Subtract $8 y'$ from both sides of the equation.

$- y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) - y' x^{2} - 8 y' = - 2 x y$

Step 5.3.5

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.5.1

Subtract $x^{4} \sec^{2} (x - y)$ from both sides of the equation.

$- y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) - y' x^{2} - 8 y' = - 2 x y - x^{4} \sec^{2} (x - y)$

Step 5.3.5.2

Subtract $16 x^{2} \sec^{2} (x - y)$ from both sides of the equation.

$- y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) + 64 \sec^{2} (x - y) - y' x^{2} - 8 y' = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y)$

Step 5.3.5.3

Subtract $64 \sec^{2} (x - y)$ from both sides of the equation.

$- y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) - y' x^{2} - 8 y' = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6

Factor $y'$ out of $- y' x^{4} \sec^{2} (x - y) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) - y' x^{2} - 8 y'$ .

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Step 5.3.6.1

Factor $y'$ out of $- y' x^{4} \sec^{2} (x - y)$ .

$y' (- 1 x^{4} \sec^{2} (x - y)) - 16 y' x^{2} \sec^{2} (x - y) - 64 y' \sec^{2} (x - y) - y' x^{2} - 8 y' = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.2

Factor $y'$ out of $- 16 y' x^{2} \sec^{2} (x - y)$ .

$y' (- 1 x^{4} \sec^{2} (x - y)) + y' (- 16 x^{2} \sec^{2} (x - y)) - 64 y' \sec^{2} (x - y) - y' x^{2} - 8 y' = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.3

Factor $y'$ out of $- 64 y' \sec^{2} (x - y)$ .

$y' (- 1 x^{4} \sec^{2} (x - y)) + y' (- 16 x^{2} \sec^{2} (x - y)) + y' (- 64 \sec^{2} (x - y)) - y' x^{2} - 8 y' = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.4

Factor $y'$ out of $- y' x^{2}$ .

$y' (- 1 x^{4} \sec^{2} (x - y)) + y' (- 16 x^{2} \sec^{2} (x - y)) + y' (- 64 \sec^{2} (x - y)) + y' (- 1 x^{2}) - 8 y' = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.5

Factor $y'$ out of $- 8 y'$ .

$y' (- 1 x^{4} \sec^{2} (x - y)) + y' (- 16 x^{2} \sec^{2} (x - y)) + y' (- 64 \sec^{2} (x - y)) + y' (- 1 x^{2}) + y' \cdot - 8 = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.6

Factor $y'$ out of $y' (- 1 x^{4} \sec^{2} (x - y)) + y' (- 16 x^{2} \sec^{2} (x - y))$ .

$y' (- 1 x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y)) + y' (- 64 \sec^{2} (x - y)) + y' (- 1 x^{2}) + y' \cdot - 8 = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.7

Factor $y'$ out of $y' (- 1 x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y)) + y' (- 64 \sec^{2} (x - y))$ .

$y' (- 1 x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)) + y' (- 1 x^{2}) + y' \cdot - 8 = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.8

Factor $y'$ out of $y' (- 1 x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)) + y' (- 1 x^{2})$ .

$y' (- 1 x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - 1 x^{2}) + y' \cdot - 8 = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.6.9

Factor $y'$ out of $y' (- 1 x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - 1 x^{2}) + y' \cdot - 8$ .

$y' (- 1 x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - 1 x^{2} - 8) = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.7

Rewrite $- 1 x^{4}$ as $- x^{4}$ .

$y' (- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - 1 x^{2} - 8) = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.8

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$y' (- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8) = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$

Step 5.3.9

Divide each term in $y' (- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8) = - 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)$ by $- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8$ and simplify.

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Step 5.3.9.1

$\frac{y' (- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} = \frac{- 2 x y}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.2

Simplify the left side.

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Step 5.3.9.2.1

Cancel the common factor of $- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8$ .

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Step 5.3.9.2.1.1

Cancel the common factor.

Step 5.3.9.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x y}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3

Simplify the right side.

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Step 5.3.9.3.1

Simplify terms.

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Step 5.3.9.3.1.1

Simplify each term.

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Step 5.3.9.3.1.1.1

Move the negative in front of the fraction.

$y' = - \frac{2 x y}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.1.2

Move the negative in front of the fraction.

$y' = - \frac{2 x y}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.1.3

Move the negative in front of the fraction.

$y' = - \frac{2 x y}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} + \frac{- 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.1.4

Move the negative in front of the fraction.

$y' = - \frac{2 x y}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2

Simplify terms.

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Step 5.3.9.3.1.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 2 x y - x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2.2

Factor $x$ out of $- 2 x y - x^{4} \sec^{2} (x - y)$ .

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Step 5.3.9.3.1.2.2.1

Factor $x$ out of $- 2 x y$ .

$y' = \frac{x (- 2 y) - x^{4} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2.2.2

Factor $x$ out of $- x^{4} \sec^{2} (x - y)$ .

$y' = \frac{x (- 2 y) + x (- x^{3} \sec^{2} (x - y))}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2.2.3

Factor $x$ out of $x (- 2 y) + x (- x^{3} \sec^{2} (x - y))$ .

$y' = \frac{x (- 2 y - x^{3} \sec^{2} (x - y))}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2.3

Combine the numerators over the common denominator.

$y' = \frac{x (- 2 y - x^{3} \sec^{2} (x - y)) - 16 x^{2} \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2.4

Factor $x$ out of $x (- 2 y - x^{3} \sec^{2} (x - y)) - 16 x^{2} \sec^{2} (x - y)$ .

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Step 5.3.9.3.1.2.4.1

Factor $x$ out of $- 16 x^{2} \sec^{2} (x - y)$ .

$y' = \frac{x (- 2 y - x^{3} \sec^{2} (x - y)) + x (- 16 x \sec^{2} (x - y))}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2.4.2

Factor $x$ out of $x (- 2 y - x^{3} \sec^{2} (x - y)) + x (- 16 x \sec^{2} (x - y))$ .

$y' = \frac{x (- 2 y - x^{3} \sec^{2} (x - y) - 16 x \sec^{2} (x - y))}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8} - \frac{64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.1.2.5

Combine the numerators over the common denominator.

$y' = \frac{x (- 2 y - x^{3} \sec^{2} (x - y) - 16 x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2

Simplify the numerator.

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Step 5.3.9.3.2.1

Apply the distributive property.

$y' = \frac{x (- 2 y) + x (- x^{3} \sec^{2} (x - y)) + x (- 16 x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.2

Simplify.

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Step 5.3.9.3.2.2.1

Rewrite using the commutative property of multiplication.

$y' = \frac{- 2 x y + x (- x^{3} \sec^{2} (x - y)) + x (- 16 x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.2.2

Rewrite using the commutative property of multiplication.

$y' = \frac{- 2 x y - x (x^{3} \sec^{2} (x - y)) + x (- 16 x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.2.3

Rewrite using the commutative property of multiplication.

$y' = \frac{- 2 x y - x (x^{3} \sec^{2} (x - y)) - 16 x (x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.3

Simplify each term.

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Step 5.3.9.3.2.3.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 5.3.9.3.2.3.1.1

Move $x^{3}$ .

$y' = \frac{- 2 x y - (x^{3} x) \sec^{2} (x - y) - 16 x (x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.3.1.2

Multiply $x^{3}$ by $x$ .

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Step 5.3.9.3.2.3.1.2.1

Raise $x$ to the power of $1$ .

$y' = \frac{- 2 x y - (x^{3} x^{1}) \sec^{2} (x - y) - 16 x (x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' = \frac{- 2 x y - x^{3 + 1} \sec^{2} (x - y) - 16 x (x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.3.1.3

Add $3$ and $1$ .

$y' = \frac{- 2 x y - x^{4} \sec^{2} (x - y) - 16 x (x \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.9.3.2.3.2.1

Move $x$ .

$y' = \frac{- 2 x y - x^{4} \sec^{2} (x - y) - 16 (x \cdot x) \sec^{2} (x - y) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.2.3.2.2

Multiply $x$ by $x$ .

$y' = \frac{- 2 x y - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3

Simplify terms.

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Step 5.3.9.3.3.1

Factor $- 1$ out of $- 2 x y$ .

$y' = \frac{- (2 x y) - x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.2

Factor $- 1$ out of $- x^{4} \sec^{2} (x - y)$ .

$y' = \frac{- (2 x y) - (x^{4} \sec^{2} (x - y)) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.3

Factor $- 1$ out of $- (2 x y) - (x^{4} \sec^{2} (x - y))$ .

$y' = \frac{- (2 x y + x^{4} \sec^{2} (x - y)) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.4

Factor $- 1$ out of $- 16 x^{2} \sec^{2} (x - y)$ .

$y' = \frac{- (2 x y + x^{4} \sec^{2} (x - y)) - (16 x^{2} \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.5

Factor $- 1$ out of $- (2 x y + x^{4} \sec^{2} (x - y)) - (16 x^{2} \sec^{2} (x - y))$ .

$y' = \frac{- (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y)) - 64 \sec^{2} (x - y)}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.6

Factor $- 1$ out of $- 64 \sec^{2} (x - y)$ .

$y' = \frac{- (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y)) - (64 \sec^{2} (x - y))}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.7

Factor $- 1$ out of $- (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y)) - (64 \sec^{2} (x - y))$ .

$y' = \frac{- (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.8

Rewrite $- (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))$ as $- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- x^{4} \sec^{2} (x - y) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.9

Factor $- 1$ out of $- x^{4} \sec^{2} (x - y)$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y)) - 16 x^{2} \sec^{2} (x - y) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.10

Factor $- 1$ out of $- 16 x^{2} \sec^{2} (x - y)$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y)) - (16 x^{2} \sec^{2} (x - y)) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.11

Factor $- 1$ out of $- (x^{4} \sec^{2} (x - y)) - (16 x^{2} \sec^{2} (x - y))$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y)) - 64 \sec^{2} (x - y) - x^{2} - 8}$

Step 5.3.9.3.3.12

Factor $- 1$ out of $- 64 \sec^{2} (x - y)$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y)) - (64 \sec^{2} (x - y)) - x^{2} - 8}$

Step 5.3.9.3.3.13

Factor $- 1$ out of $- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y)) - (64 \sec^{2} (x - y))$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y)) - x^{2} - 8}$

Step 5.3.9.3.3.14

Factor $- 1$ out of $- x^{2}$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y)) - (x^{2}) - 8}$

Step 5.3.9.3.3.15

Factor $- 1$ out of $- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y)) - (x^{2})$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2}) - 8}$

Step 5.3.9.3.3.16

Rewrite $- 8$ as $- 1 (8)$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2}) - 1 (8)}$

Step 5.3.9.3.3.17

Factor $- 1$ out of $- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2}) - 1 (8)$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2} + 8)}$

Step 5.3.9.3.3.18

Rewrite $- (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2} + 8)$ as $- 1 (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2} + 8)$ .

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- 1 (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2} + 8)}$

Step 5.3.9.3.3.19

Cancel the common factor.

$y' = \frac{- 1 (2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y))}{- 1 (x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2} + 8)}$

Step 5.3.9.3.3.20

Rewrite the expression.

$y' = \frac{2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2} + 8}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2 x y + x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y)}{x^{4} \sec^{2} (x - y) + 16 x^{2} \sec^{2} (x - y) + 64 \sec^{2} (x - y) + x^{2} + 8}$