Calculus Examples

$2 {(x^{2} + y^{2})}^{2} = 25 x y^{2}$ ; $(2, 1)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (2 {(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (25 x y^{2})$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 {(x^{2} + y^{2})}^{2}$ with respect to $x$ is $2 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

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

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + y^{2}$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

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

Step 1.2.3

Differentiate.

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Step 1.2.3.1

Multiply $2$ by $2$ .

$4 ((x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Step 1.2.3.2

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

$4 (x^{2} + y^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

Step 1.2.3.3

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

$4 (x^{2} + y^{2}) (2 x + \frac{d}{d x} [y^{2}])$

Step 1.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 1.2.4.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$4 (x^{2} + y^{2}) (2 x + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 1.2.4.2

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

$4 (x^{2} + y^{2}) (2 x + 2 u_{2} \frac{d}{d x} [y])$

Step 1.2.4.3

Replace all occurrences of $u_{2}$ with $y$ .

$4 (x^{2} + y^{2}) (2 x + 2 y \frac{d}{d x} [y])$

Step 1.2.5

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

$4 (x^{2} + y^{2}) (2 x + 2 y y')$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Apply the distributive property.

$(4 x^{2} + 4 y^{2}) (2 x + 2 y y')$

Step 1.2.6.2

Reorder the factors of $(4 x^{2} + 4 y^{2}) (2 x + 2 y y')$ .

$(2 x + 2 y y') (4 x^{2} + 4 y^{2})$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Since $25$ is constant with respect to $x$ , the derivative of $25 x y^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x y^{2}]$ .

$25 \frac{d}{d x} [x y^{2}]$

Step 1.3.2

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

$25 (x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x])$

Step 1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 1.3.3.1

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

$25 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.3.2

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

$25 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.3.3

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

$25 (x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.4

Move $2$ to the left of $x$ .

$25 (2 \cdot x y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x])$

Step 1.3.5

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

$25 (2 x y y' + y^{2} \frac{d}{d x} [x])$

Step 1.3.6

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

$25 (2 x y y' + y^{2} \cdot 1)$

Step 1.3.7

Multiply $y^{2}$ by $1$ .

$25 (2 x y y' + y^{2})$

Step 1.3.8

Simplify.

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Step 1.3.8.1

Apply the distributive property.

$25 (2 x y y') + 25 y^{2}$

Step 1.3.8.2

Multiply $2$ by $25$ .

$50 x y y' + 25 y^{2}$

Step 1.3.8.3

Reorder terms.

$25 y^{2} + 50 x y y'$

Step 1.4

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

$(2 x + 2 y y') (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Simplify $(2 x + 2 y y') (4 x^{2} + 4 y^{2})$ .

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Step 1.5.1.1

Rewrite.

$0 + 0 + (2 x + 2 y y') (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.3

Expand $(2 x + 2 y y') (4 x^{2} + 4 y^{2})$ using the FOIL Method.

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Step 1.5.1.3.1

Apply the distributive property.

$2 x (4 x^{2} + 4 y^{2}) + 2 y y' (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.3.2

Apply the distributive property.

$2 x (4 x^{2}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.3.3

Apply the distributive property.

$2 x (4 x^{2}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4

Simplify each term.

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Step 1.5.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 4 x \cdot x^{2} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2

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

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Step 1.5.1.4.2.1

Move $x^{2}$ .

$2 \cdot 4 (x^{2} x) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2.2

Multiply $x^{2}$ by $x$ .

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Step 1.5.1.4.2.2.1

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

$2 \cdot 4 (x^{2} x^{1}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2.2.2

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

$2 \cdot 4 x^{2 + 1} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2.3

Add $2$ and $1$ .

$2 \cdot 4 x^{3} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.3

Multiply $2$ by $4$ .

$8 x^{3} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.4

Rewrite using the commutative property of multiplication.

$8 x^{3} + 2 \cdot 4 x y^{2} + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.5

Multiply $2$ by $4$ .

$8 x^{3} + 8 x y^{2} + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.6

Multiply $4$ by $2$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7

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

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Step 1.5.1.4.7.1

Move $y^{2}$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 (y^{2} y) y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7.2

Multiply $y^{2}$ by $y$ .

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Step 1.5.1.4.7.2.1

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

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7.2.2

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

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y^{2 + 1} y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7.3

Add $2$ and $1$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y^{3} y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.8

Multiply $4$ by $2$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' = 25 y^{2} + 50 x y y'$

Step 1.5.2

Subtract $50 x y y'$ from both sides of the equation.

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' - 50 x y y' = 25 y^{2}$

Step 1.5.3

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

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Step 1.5.3.1

Subtract $8 x^{3}$ from both sides of the equation.

$8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' - 50 x y y' = 25 y^{2} - 8 x^{3}$

Step 1.5.3.2

Subtract $8 x y^{2}$ from both sides of the equation.

$8 y y' x^{2} + 8 y^{3} y' - 50 x y y' = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4

Factor $2 y y'$ out of $8 y y' x^{2} + 8 y^{3} y' - 50 x y y'$ .

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Step 1.5.4.1

Factor $2 y y'$ out of $8 y y' x^{2}$ .

$2 y y' (4 x^{2}) + 8 y^{3} y' - 50 x y y' = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.2

Factor $2 y y'$ out of $8 y^{3} y'$ .

$2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) - 50 x y y' = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.3

Factor $2 y y'$ out of $- 50 x y y'$ .

$2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) + 2 y y' (- 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.4

Factor $2 y y'$ out of $2 y y' (4 x^{2}) + 2 y y' (4 y^{2})$ .

$2 y y' (4 x^{2} + 4 y^{2}) + 2 y y' (- 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.5

Factor $2 y y'$ out of $2 y y' (4 x^{2} + 4 y^{2}) + 2 y y' (- 25 x)$ .

$2 y y' (4 x^{2} + 4 y^{2} - 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.5

Divide each term in $2 y y' (4 x^{2} + 4 y^{2} - 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$ by $2 y (4 x^{2} + 4 y^{2} - 25 x)$ and simplify.

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Step 1.5.5.1

Divide each term in $2 y y' (4 x^{2} + 4 y^{2} - 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$ by $2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$\frac{2 y y' (4 x^{2} + 4 y^{2} - 25 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)} = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.2

Simplify the left side.

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Step 1.5.5.2.1

Cancel the common factor of $2$ .

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Step 1.5.5.2.1.1

Cancel the common factor.

Step 1.5.5.2.1.2

Rewrite the expression.

$\frac{y y' (4 x^{2} + 4 y^{2} - 25 x)}{y (4 x^{2} + 4 y^{2} - 25 x)} = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.2.2

Cancel the common factor of $y$ .

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Step 1.5.5.2.2.1

Cancel the common factor.

Step 1.5.5.2.2.2

Rewrite the expression.

$\frac{y' (4 x^{2} + 4 y^{2} - 25 x)}{4 x^{2} + 4 y^{2} - 25 x} = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.2.3

Cancel the common factor of $4 x^{2} + 4 y^{2} - 25 x$ .

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Step 1.5.5.2.3.1

Cancel the common factor.

Step 1.5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3

Simplify the right side.

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Step 1.5.5.3.1

Simplify each term.

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Step 1.5.5.3.1.1

Cancel the common factor of $y^{2}$ and $y$ .

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Step 1.5.5.3.1.1.1

Factor $y$ out of $25 y^{2}$ .

$y' = \frac{y (25 y)}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.1.2

Cancel the common factors.

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Step 1.5.5.3.1.1.2.1

Factor $y$ out of $2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$y' = \frac{y (25 y)}{y (2 (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{y (25 y)}{y (2 (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2

Cancel the common factor of $- 8$ and $2$ .

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Step 1.5.5.3.1.2.1

Factor $2$ out of $- 8 x^{3}$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x^{3})}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2.2

Cancel the common factors.

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Step 1.5.5.3.1.2.2.1

Factor $2$ out of $2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x^{3})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2.2.2

Cancel the common factor.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x^{3})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2.2.3

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.4

Cancel the common factor of $- 8$ and $2$ .

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Step 1.5.5.3.1.4.1

Factor $2$ out of $- 8 x y^{2}$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x y^{2})}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.4.2

Cancel the common factors.

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Step 1.5.5.3.1.4.2.1

Factor $2$ out of $2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x y^{2})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))}$

Step 1.5.5.3.1.4.2.2

Cancel the common factor.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x y^{2})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))}$

Step 1.5.5.3.1.4.2.3

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 4 x y^{2}}{y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.5

Cancel the common factor of $y^{2}$ and $y$ .

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Step 1.5.5.3.1.5.1

Factor $y$ out of $- 4 x y^{2}$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (- 4 x y)}{y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.5.2

Cancel the common factors.

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Step 1.5.5.3.1.5.2.1

Cancel the common factor.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (- 4 x y)}{y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.5.2.2

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 4 x y}{4 x^{2} + 4 y^{2} - 25 x}$

Step 1.5.5.3.1.6

Move the negative in front of the fraction.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x}$

Step 1.5.5.3.2

To write $- \frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x} \cdot \frac{2}{2}$

Step 1.5.5.3.3

Write each expression with a common denominator of $2 (4 x^{2} + 4 y^{2} - 25 x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.5.3.3.1

Multiply $\frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x}$ by $\frac{2}{2}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y \cdot 2}{(4 x^{2} + 4 y^{2} - 25 x) \cdot 2}$

Step 1.5.5.3.3.2

Reorder the factors of $(4 x^{2} + 4 y^{2} - 25 x) \cdot 2$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y \cdot 2}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.4

Combine the numerators over the common denominator.

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y - 4 x y \cdot 2}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5

Simplify the numerator.

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Step 1.5.5.3.5.1

Factor $y$ out of $25 y - 4 x y \cdot 2$ .

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Step 1.5.5.3.5.1.1

Factor $y$ out of $25 y$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y \cdot 25 - 4 x y \cdot 2}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5.1.2

Factor $y$ out of $- 4 x y \cdot 2$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y \cdot 25 + y (- 4 x \cdot 2)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5.1.3

Factor $y$ out of $y \cdot 25 + y (- 4 x \cdot 2)$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 4 x \cdot 2)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5.2

Multiply $2$ by $- 4$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.6

To write $- \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{2}{2} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.7

To write $\frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{2}{2} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{y}{y}$

Step 1.5.5.3.8

Write each expression with a common denominator of $y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.5.3.8.1

Multiply $\frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)}$ by $\frac{2}{2}$ .

$y' = - \frac{4 x^{3} \cdot 2}{y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{y}{y}$

Step 1.5.5.3.8.2

Multiply $\frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$ by $\frac{y}{y}$ .

$y' = - \frac{4 x^{3} \cdot 2}{y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2} + \frac{y (25 - 8 x) y}{2 (4 x^{2} + 4 y^{2} - 25 x) y}$

Step 1.5.5.3.8.3

Reorder the factors of $y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2$ .

$y' = - \frac{4 x^{3} \cdot 2}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 8 x) y}{2 (4 x^{2} + 4 y^{2} - 25 x) y}$

Step 1.5.5.3.8.4

Reorder the factors of $2 (4 x^{2} + 4 y^{2} - 25 x) y$ .

$y' = - \frac{4 x^{3} \cdot 2}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 8 x) y}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.9

Combine the numerators over the common denominator.

$y' = \frac{- 4 x^{3} \cdot 2 + y (25 - 8 x) y}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10

Simplify the numerator.

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Step 1.5.5.3.10.1

Multiply $2$ by $- 4$ .

$y' = \frac{- 8 x^{3} + y (25 - 8 x) y}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.5.5.3.10.2.1

Move $y$ .

$y' = \frac{- 8 x^{3} + y \cdot y (25 - 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.2.2

Multiply $y$ by $y$ .

$y' = \frac{- 8 x^{3} + y^{2} (25 - 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.3

Apply the distributive property.

$y' = \frac{- 8 x^{3} + y^{2} \cdot 25 + y^{2} (- 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.4

Move $25$ to the left of $y^{2}$ .

$y' = \frac{- 8 x^{3} + 25 \cdot y^{2} + y^{2} (- 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.5

Rewrite using the commutative property of multiplication.

$y' = \frac{- 8 x^{3} + 25 y^{2} - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11

Simplify with factoring out.

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Step 1.5.5.3.11.1

Factor $- 1$ out of $- 8 x^{3}$ .

$y' = \frac{- (8 x^{3}) + 25 y^{2} - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.2

Factor $- 1$ out of $25 y^{2}$ .

$y' = \frac{- (8 x^{3}) - (- 25 y^{2}) - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.3

Factor $- 1$ out of $- (8 x^{3}) - (- 25 y^{2})$ .

$y' = \frac{- (8 x^{3} - 25 y^{2}) - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.4

Factor $- 1$ out of $- 8 y^{2} x$ .

$y' = \frac{- (8 x^{3} - 25 y^{2}) - (8 y^{2} x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.5

Factor $- 1$ out of $- (8 x^{3} - 25 y^{2}) - (8 y^{2} x)$ .

$y' = \frac{- (8 x^{3} - 25 y^{2} + 8 y^{2} x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.6

Simplify the expression.

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Step 1.5.5.3.11.6.1

Rewrite $- (8 x^{3} - 25 y^{2} + 8 y^{2} x)$ as $- 1 (8 x^{3} - 25 y^{2} + 8 y^{2} x)$ .

$y' = \frac{- 1 (8 x^{3} - 25 y^{2} + 8 y^{2} x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.6.2

Move the negative in front of the fraction.

$y' = - \frac{8 x^{3} - 25 y^{2} + 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{8 x^{3} - 25 y^{2} + 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.7

Evaluate at $x = 2$ and $y = 1$ .

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Step 1.7.1

Replace the variable $x$ with $2$ in the expression.

$- \frac{8 {(2)}^{3} - 25 y^{2} + 8 y^{2} (2)}{2 y (4 {(2)}^{2} + 4 y^{2} - 25 \cdot 2)}$

Step 1.7.2

Replace the variable $y$ with $1$ in the expression.

$- \frac{8 {(2)}^{3} - 25 {(1)}^{2} + 8 {(1)}^{2} (2)}{2 (1) (4 {(2)}^{2} + 4 {(1)}^{2} - 25 \cdot 2)}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Raise $2$ to the power of $3$ .

$- \frac{8 \cdot 8 - 25 \cdot 1^{2} + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.2

Multiply $8$ by $8$ .

$- \frac{64 - 25 \cdot 1^{2} + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.3

One to any power is one.

$- \frac{64 - 25 \cdot 1 + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.4

Multiply $- 25$ by $1$ .

$- \frac{64 - 25 + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.5

One to any power is one.

$- \frac{64 - 25 + 8 \cdot 1 \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.6

Multiply $8 \cdot 1 \cdot 2$ .

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Step 1.7.3.6.1

Multiply $8$ by $1$ .

$- \frac{64 - 25 + 8 \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.6.2

Multiply $8$ by $2$ .

$- \frac{64 - 25 + 16}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.7

Subtract $25$ from $64$ .

$- \frac{39 + 16}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.8

Add $39$ and $16$ .

$- \frac{55}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.4

Multiply $2$ by $1$ .

$- \frac{55}{2 (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Raise $2$ to the power of $2$ .

$- \frac{55}{2 (4 \cdot 4 + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.5.2

Multiply $4$ by $4$ .

$- \frac{55}{2 (16 + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.5.3

One to any power is one.

$- \frac{55}{2 (16 + 4 \cdot 1 - 25 \cdot 2)}$

Step 1.7.5.4

Multiply $4$ by $1$ .

$- \frac{55}{2 (16 + 4 - 25 \cdot 2)}$

Step 1.7.5.5

Multiply $- 25$ by $2$ .

$- \frac{55}{2 (16 + 4 - 50)}$

Step 1.7.5.6

Add $16$ and $4$ .

$- \frac{55}{2 (20 - 50)}$

Step 1.7.5.7

Subtract $50$ from $20$ .

$- \frac{55}{2 \cdot - 30}$

Step 1.7.6

Reduce the expression by cancelling the common factors.

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Step 1.7.6.1

Multiply $2$ by $- 30$ .

$- \frac{55}{- 60}$

Step 1.7.6.2

Cancel the common factor of $55$ and $- 60$ .

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Step 1.7.6.2.1

Factor $5$ out of $55$ .

$- \frac{5 (11)}{- 60}$

Step 1.7.6.2.2

Cancel the common factors.

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Step 1.7.6.2.2.1

Factor $5$ out of $- 60$ .

$- \frac{5 \cdot 11}{5 \cdot - 12}$

Step 1.7.6.2.2.2

Cancel the common factor.

$- \frac{5 \cdot 11}{5 \cdot - 12}$

Step 1.7.6.2.2.3

Rewrite the expression.

$- \frac{11}{- 12}$

Step 1.7.6.3

Move the negative in front of the fraction.

$- - \frac{11}{12}$

Step 1.7.7

Multiply $- - \frac{11}{12}$ .

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Step 1.7.7.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{11}{12})$

Step 1.7.7.2

Multiply $\frac{11}{12}$ by $1$ .

$\frac{11}{12}$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $\frac{11}{12}$ and a given point $(2, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{11}{12} \cdot (x - (2))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 1 = \frac{11}{12} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{11}{12} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{11}{12} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = \frac{11}{12} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = \frac{11}{12} x + \frac{11}{12} \cdot - 2$

Step 2.3.1.4

Combine $\frac{11}{12}$ and $x$ .

$y - 1 = \frac{11 x}{12} + \frac{11}{12} \cdot - 2$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Factor $2$ out of $12$ .

$y - 1 = \frac{11 x}{12} + \frac{11}{2 (6)} \cdot - 2$

Step 2.3.1.5.2

Factor $2$ out of $- 2$ .

$y - 1 = \frac{11 x}{12} + \frac{11}{2 \cdot 6} \cdot (2 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 1 = \frac{11 x}{12} + \frac{11}{2 \cdot 6} \cdot (2 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 1 = \frac{11 x}{12} + \frac{11}{6} \cdot - 1$

Step 2.3.1.6

Combine $\frac{11}{6}$ and $- 1$ .

$y - 1 = \frac{11 x}{12} + \frac{11 \cdot - 1}{6}$

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply $11$ by $- 1$ .

$y - 1 = \frac{11 x}{12} + \frac{- 11}{6}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - 1 = \frac{11 x}{12} - \frac{11}{6}$

Step 2.3.2

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

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Step 2.3.2.1

Add $1$ to both sides of the equation.

$y = \frac{11 x}{12} - \frac{11}{6} + 1$

Step 2.3.2.2

Write $1$ as a fraction with a common denominator.

$y = \frac{11 x}{12} - \frac{11}{6} + \frac{6}{6}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$y = \frac{11 x}{12} + \frac{- 11 + 6}{6}$

Step 2.3.2.4

Add $- 11$ and $6$ .

$y = \frac{11 x}{12} + \frac{- 5}{6}$

Step 2.3.2.5

Move the negative in front of the fraction.

$y = \frac{11 x}{12} - \frac{5}{6}$

Step 2.3.3

Reorder terms.

$y = \frac{11}{12} x - \frac{5}{6}$

Step 3