Calculus Examples

$4 x^{2} y^{7} - 2 x = x^{5} + 4 y^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (4 x^{2} y^{7} - 2 x) = \frac{d}{d x} (x^{5} + 4 y^{3})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{2} y^{7} - 2 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{2} y^{7}] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [4 x^{2} y^{7}] + \frac{d}{d x} [- 2 x]$

Step 2.2

Evaluate $\frac{d}{d x} [4 x^{2} y^{7}]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2} y^{7}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2} y^{7}]$ .

$4 \frac{d}{d x} [x^{2} y^{7}] + \frac{d}{d x} [- 2 x]$

Step 2.2.2

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 $f (x) = x^{2}$ and $g (x) = y^{7}$ .

$4 (x^{2} \frac{d}{d x} [y^{7}] + y^{7} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 2 x]$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = y$ .

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Step 2.2.3.1

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

$4 (x^{2} (\frac{d}{d u} [u^{7}] \frac{d}{d x} [y]) + y^{7} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 2 x]$

Step 2.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$4 (x^{2} (7 u^{6} \frac{d}{d x} [y]) + y^{7} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 2 x]$

Step 2.2.3.3

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

$4 (x^{2} (7 y^{6} \frac{d}{d x} [y]) + y^{7} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 2 x]$

Step 2.2.4

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

$4 (x^{2} (7 y^{6} y') + y^{7} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 2 x]$

Step 2.2.5

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

$4 (x^{2} (7 y^{6} y') + y^{7} (2 x)) + \frac{d}{d x} [- 2 x]$

Step 2.2.6

Move $7$ to the left of $x^{2}$ .

$4 (7 \cdot x^{2} y^{6} y' + y^{7} (2 x)) + \frac{d}{d x} [- 2 x]$

Step 2.2.7

Move $2$ to the left of $y^{7}$ .

$4 (7 x^{2} y^{6} y' + 2 y^{7} x) + \frac{d}{d x} [- 2 x]$

Step 2.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.3.1

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

$4 (7 x^{2} y^{6} y' + 2 y^{7} x) - 2 \frac{d}{d x} [x]$

Step 2.3.2

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

$4 (7 x^{2} y^{6} y' + 2 y^{7} x) - 2 \cdot 1$

Step 2.3.3

Multiply $- 2$ by $1$ .

$4 (7 x^{2} y^{6} y' + 2 y^{7} x) - 2$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$4 (7 x^{2} y^{6} y') + 4 (2 y^{7} x) - 2$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $7$ by $4$ .

$28 (x^{2} y^{6} y') + 4 (2 y^{7} x) - 2$

Step 2.4.2.2

Multiply $2$ by $4$ .

$28 x^{2} y^{6} y' + 8 y^{7} x - 2$

Step 2.4.3

Reorder terms.

$28 y^{6} x^{2} y' + 8 y^{7} x - 2$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $x^{5} + 4 y^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [4 y^{3}]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [4 y^{3}]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [4 y^{3}]$

Step 3.2

Evaluate $\frac{d}{d x} [4 y^{3}]$ .

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 y^{3}$ with respect to $x$ is $4 \frac{d}{d x} [y^{3}]$ .

$5 x^{4} + 4 \frac{d}{d x} [y^{3}]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 3.2.2.1

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

$5 x^{4} + 4 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y])$

Step 3.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$5 x^{4} + 4 (3 u^{2} \frac{d}{d x} [y])$

Step 3.2.2.3

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

$5 x^{4} + 4 (3 y^{2} \frac{d}{d x} [y])$

Step 3.2.3

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

$5 x^{4} + 4 (3 y^{2} y')$

Step 3.2.4

Multiply $3$ by $4$ .

$5 x^{4} + 12 y^{2} y'$

Step 4

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

$28 y^{6} x^{2} y' + 8 y^{7} x - 2 = 5 x^{4} + 12 y^{2} y'$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $12 y^{2} y'$ from both sides of the equation.

$28 y^{6} x^{2} y' + 8 y^{7} x - 2 - 12 y^{2} y' = 5 x^{4}$

Step 5.2

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

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Step 5.2.1

Subtract $8 y^{7} x$ from both sides of the equation.

$28 y^{6} x^{2} y' - 2 - 12 y^{2} y' = 5 x^{4} - 8 y^{7} x$

Step 5.2.2

Add $2$ to both sides of the equation.

$28 y^{6} x^{2} y' - 12 y^{2} y' = 5 x^{4} - 8 y^{7} x + 2$

Step 5.3

Factor $4 y^{2} y'$ out of $28 y^{6} x^{2} y' - 12 y^{2} y'$ .

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Step 5.3.1

Factor $4 y^{2} y'$ out of $28 y^{6} x^{2} y'$ .

$4 y^{2} y' (7 y^{4} x^{2}) - 12 y^{2} y' = 5 x^{4} - 8 y^{7} x + 2$

Step 5.3.2

Factor $4 y^{2} y'$ out of $- 12 y^{2} y'$ .

$4 y^{2} y' (7 y^{4} x^{2}) + 4 y^{2} y' \cdot - 3 = 5 x^{4} - 8 y^{7} x + 2$

Step 5.3.3

Factor $4 y^{2} y'$ out of $4 y^{2} y' (7 y^{4} x^{2}) + 4 y^{2} y' \cdot - 3$ .

$4 y^{2} y' (7 y^{4} x^{2} - 3) = 5 x^{4} - 8 y^{7} x + 2$

Step 5.4

Divide each term in $4 y^{2} y' (7 y^{4} x^{2} - 3) = 5 x^{4} - 8 y^{7} x + 2$ by $4 y^{2} (7 y^{4} x^{2} - 3)$ and simplify.

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Step 5.4.1

Divide each term in $4 y^{2} y' (7 y^{4} x^{2} - 3) = 5 x^{4} - 8 y^{7} x + 2$ by $4 y^{2} (7 y^{4} x^{2} - 3)$ .

$\frac{4 y^{2} y' (7 y^{4} x^{2} - 3)}{4 y^{2} (7 y^{4} x^{2} - 3)} = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{- 8 y^{7} x}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $4$ .

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Step 5.4.2.1.1

Cancel the common factor.

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y^{2} y' (7 y^{4} x^{2} - 3)}{y^{2} (7 y^{4} x^{2} - 3)} = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{- 8 y^{7} x}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.2.2

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

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Step 5.4.2.2.1

Cancel the common factor.

Step 5.4.2.2.2

Rewrite the expression.

$\frac{y' (7 y^{4} x^{2} - 3)}{7 y^{4} x^{2} - 3} = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{- 8 y^{7} x}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.2.3

Cancel the common factor of $7 y^{4} x^{2} - 3$ .

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Step 5.4.2.3.1

Cancel the common factor.

$\frac{y' (7 y^{4} x^{2} - 3)}{7 y^{4} x^{2} - 3} = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{- 8 y^{7} x}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{- 8 y^{7} x}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $- 8$ and $4$ .

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Step 5.4.3.1.1.1

Factor $4$ out of $- 8 y^{7} x$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{4 (- 2 y^{7} x)}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $4$ out of $4 y^{2} (7 y^{4} x^{2} - 3)$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{4 (- 2 y^{7} x)}{4 (y^{2} (7 y^{4} x^{2} - 3))} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{4 (- 2 y^{7} x)}{4 (y^{2} (7 y^{4} x^{2} - 3))} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{- 2 y^{7} x}{y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.2

Cancel the common factor of $y^{7}$ and $y^{2}$ .

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Step 5.4.3.1.2.1

Factor $y^{2}$ out of $- 2 y^{7} x$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{y^{2} (- 2 y^{5} x)}{y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.2.2

Cancel the common factors.

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Step 5.4.3.1.2.2.1

Cancel the common factor.

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{y^{2} (- 2 y^{5} x)}{y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.2.2.2

Rewrite the expression.

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{- 2 y^{5} x}{7 y^{4} x^{2} - 3} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.3

Move the negative in front of the fraction.

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{(2) y^{5} x}{7 y^{4} x^{2} - 3} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.4

Cancel the common factor of $2$ and $4$ .

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Step 5.4.3.1.4.1

Factor $2$ out of $2$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{2 y^{5} x}{7 y^{4} x^{2} - 3} + \frac{2 (1)}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.1.4.2

Cancel the common factors.

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Step 5.4.3.1.4.2.1

Factor $2$ out of $4 y^{2} (7 y^{4} x^{2} - 3)$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{2 y^{5} x}{7 y^{4} x^{2} - 3} + \frac{2 (1)}{2 (2 y^{2} (7 y^{4} x^{2} - 3))}$

Step 5.4.3.1.4.2.2

Cancel the common factor.

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{2 y^{5} x}{7 y^{4} x^{2} - 3} + \frac{2 \cdot 1}{2 (2 y^{2} (7 y^{4} x^{2} - 3))}$

Step 5.4.3.1.4.2.3

Rewrite the expression.

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{2 y^{5} x}{7 y^{4} x^{2} - 3} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.2

To write $- \frac{2 y^{5} x}{7 y^{4} x^{2} - 3}$ as a fraction with a common denominator, multiply by $\frac{4 y^{2}}{4 y^{2}}$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{2 y^{5} x}{7 y^{4} x^{2} - 3} \cdot \frac{4 y^{2}}{4 y^{2}} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.3

Write each expression with a common denominator of $4 (7 y^{4} x^{2} - 3) y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.1

Multiply $\frac{2 y^{5} x}{7 y^{4} x^{2} - 3}$ by $\frac{4 y^{2}}{4 y^{2}}$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{2 y^{5} x (4 y^{2})}{(7 y^{4} x^{2} - 3) (4 y^{2})} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.3.2

Reorder the factors of $(7 y^{4} x^{2} - 3) (4 y^{2})$ .

$y' = \frac{5 x^{4}}{4 y^{2} (7 y^{4} x^{2} - 3)} - \frac{2 y^{5} x (4 y^{2})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.4

Combine the numerators over the common denominator.

$y' = \frac{5 x^{4} - 2 y^{5} x (4 y^{2})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5

Simplify the numerator.

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Step 5.4.3.5.1

Factor $x$ out of $5 x^{4} - 2 y^{5} x (4 y^{2})$ .

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Step 5.4.3.5.1.1

Factor $x$ out of $5 x^{4}$ .

$y' = \frac{x (5 x^{3}) - 2 y^{5} x (4 y^{2})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5.1.2

Factor $x$ out of $- 2 y^{5} x (4 y^{2})$ .

$y' = \frac{x (5 x^{3}) + x (- 2 y^{5} (4 y^{2}))}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5.1.3

Factor $x$ out of $x (5 x^{3}) + x (- 2 y^{5} (4 y^{2}))$ .

$y' = \frac{x (5 x^{3} - 2 y^{5} (4 y^{2}))}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5.2

Rewrite using the commutative property of multiplication.

$y' = \frac{x (5 x^{3} - 2 \cdot 4 y^{5} y^{2})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5.3

Multiply $y^{5}$ by $y^{2}$ by adding the exponents.

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Step 5.4.3.5.3.1

Move $y^{2}$ .

$y' = \frac{x (5 x^{3} - 2 \cdot 4 (y^{2} y^{5}))}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5.3.2

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

$y' = \frac{x (5 x^{3} - 2 \cdot 4 y^{2 + 5})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5.3.3

Add $2$ and $5$ .

$y' = \frac{x (5 x^{3} - 2 \cdot 4 y^{7})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.5.4

Multiply $- 2$ by $4$ .

$y' = \frac{x (5 x^{3} - 8 y^{7})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.6

To write $\frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = \frac{x (5 x^{3} - 8 y^{7})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)} \cdot \frac{2}{2}$

Step 5.4.3.7

Write each expression with a common denominator of $4 (7 y^{4} x^{2} - 3) y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.7.1

Multiply $\frac{1}{2 y^{2} (7 y^{4} x^{2} - 3)}$ by $\frac{2}{2}$ .

$y' = \frac{x (5 x^{3} - 8 y^{7})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{2 y^{2} (7 y^{4} x^{2} - 3) \cdot 2}$

Step 5.4.3.7.2

Multiply $2$ by $2$ .

$y' = \frac{x (5 x^{3} - 8 y^{7})}{4 y^{2} (7 y^{4} x^{2} - 3)} + \frac{2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.8

Combine the numerators over the common denominator.

$y' = \frac{x (5 x^{3} - 8 y^{7}) + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.9

Simplify the numerator.

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Step 5.4.3.9.1

Apply the distributive property.

$y' = \frac{x (5 x^{3}) + x (- 8 y^{7}) + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.9.2

Rewrite using the commutative property of multiplication.

$y' = \frac{5 x \cdot x^{3} + x (- 8 y^{7}) + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.9.3

Rewrite using the commutative property of multiplication.

$y' = \frac{5 x \cdot x^{3} - 8 x y^{7} + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.9.4

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

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Step 5.4.3.9.4.1

Move $x^{3}$ .

$y' = \frac{5 (x^{3} x) - 8 x y^{7} + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.9.4.2

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

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Step 5.4.3.9.4.2.1

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

$y' = \frac{5 (x^{3} x^{1}) - 8 x y^{7} + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.9.4.2.2

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

$y' = \frac{5 x^{3 + 1} - 8 x y^{7} + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 5.4.3.9.4.3

Add $3$ and $1$ .

$y' = \frac{5 x^{4} - 8 x y^{7} + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$

Step 6

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

$\frac{d y}{d x} = \frac{5 x^{4} - 8 x y^{7} + 2}{4 y^{2} (7 y^{4} x^{2} - 3)}$