Calculus Examples

$x = \frac{y^{4}}{4} + \frac{1}{8 y^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x) = \frac{d}{d y} (\frac{y^{4}}{4} + \frac{1}{8 y^{2}})$

Step 2

The derivative of $x$ with respect to $y$ is $x'$ .

$x'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $\frac{y^{4}}{4} + \frac{1}{8 y^{2}}$ with respect to $y$ is $\frac{d}{d y} [\frac{y^{4}}{4}] + \frac{d}{d y} [\frac{1}{8 y^{2}}]$ .

$\frac{d}{d y} [\frac{y^{4}}{4}] + \frac{d}{d y} [\frac{1}{8 y^{2}}]$

Step 3.2

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

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Step 3.2.1

Since $\frac{1}{4}$ is constant with respect to $y$ , the derivative of $\frac{y^{4}}{4}$ with respect to $y$ is $\frac{1}{4} \frac{d}{d y} [y^{4}]$ .

$\frac{1}{4} \frac{d}{d y} [y^{4}] + \frac{d}{d y} [\frac{1}{8 y^{2}}]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$\frac{1}{4} (4 y^{3}) + \frac{d}{d y} [\frac{1}{8 y^{2}}]$

Step 3.2.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} y^{3} + \frac{d}{d y} [\frac{1}{8 y^{2}}]$

Step 3.2.4

Combine $\frac{4}{4}$ and $y^{3}$ .

$\frac{4 y^{3}}{4} + \frac{d}{d y} [\frac{1}{8 y^{2}}]$

Step 3.2.5

Cancel the common factor of $4$ .

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Step 3.2.5.1

Cancel the common factor.

$\frac{4 y^{3}}{4} + \frac{d}{d y} [\frac{1}{8 y^{2}}]$

Step 3.2.5.2

Divide $y^{3}$ by $1$ .

$y^{3} + \frac{d}{d y} [\frac{1}{8 y^{2}}]$

Step 3.3

Evaluate $\frac{d}{d y} [\frac{1}{8 y^{2}}]$ .

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Step 3.3.1

Since $\frac{1}{8}$ is constant with respect to $y$ , the derivative of $\frac{1}{8 y^{2}}$ with respect to $y$ is $\frac{1}{8} \frac{d}{d y} [\frac{1}{y^{2}}]$ .

$y^{3} + \frac{1}{8} \frac{d}{d y} [\frac{1}{y^{2}}]$

Step 3.3.2

Rewrite $\frac{1}{y^{2}}$ as ${(y^{2})}^{- 1}$ .

$y^{3} + \frac{1}{8} \frac{d}{d y} [{(y^{2})}^{- 1}]$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 1}$ and $g (y) = y^{2}$ .

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Step 3.3.3.1

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

$y^{3} + \frac{1}{8} (\frac{d}{d u} [u^{- 1}] \frac{d}{d y} [y^{2}])$

Step 3.3.3.2

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

$y^{3} + \frac{1}{8} (- u^{- 2} \frac{d}{d y} [y^{2}])$

Step 3.3.3.3

Replace all occurrences of $u$ with $y^{2}$ .

$y^{3} + \frac{1}{8} (- {(y^{2})}^{- 2} \frac{d}{d y} [y^{2}])$

Step 3.3.4

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

$y^{3} + \frac{1}{8} (- {(y^{2})}^{- 2} (2 y))$

Step 3.3.5

Multiply the exponents in ${(y^{2})}^{- 2}$ .

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Step 3.3.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.3.5.2

Multiply $2$ by $- 2$ .

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

Step 3.3.6

Multiply $2$ by $- 1$ .

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

Step 3.3.7

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

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

Step 3.3.8

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

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

Step 3.3.9

Subtract $4$ from $1$ .

$y^{3} + \frac{1}{8} (- 2 y^{- 3})$

Step 3.3.10

Combine $- 2$ and $\frac{1}{8}$ .

$y^{3} + \frac{- 2}{8} y^{- 3}$

Step 3.3.11

Combine $\frac{- 2}{8}$ and $y^{- 3}$ .

$y^{3} + \frac{- 2 y^{- 3}}{8}$

Step 3.3.12

Move $y^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y^{3} + \frac{- 2}{8 y^{3}}$

Step 3.3.13

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

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Step 3.3.13.1

Factor $2$ out of $- 2$ .

$y^{3} + \frac{2 (- 1)}{8 y^{3}}$

Step 3.3.13.2

Cancel the common factors.

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Step 3.3.13.2.1

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

$y^{3} + \frac{2 (- 1)}{2 (4 y^{3})}$

Step 3.3.13.2.2

Cancel the common factor.

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

Step 3.3.13.2.3

Rewrite the expression.

$y^{3} + \frac{- 1}{4 y^{3}}$

Step 3.3.14

Move the negative in front of the fraction.

$y^{3} - \frac{1}{4 y^{3}}$

Step 4

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

$x' = y^{3} - \frac{1}{4 y^{3}}$

Step 5

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

$\frac{d x}{d y} = y^{3} - \frac{1}{4 y^{3}}$