Calculus Examples

$x^{4} + y^{4} = 1$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = y$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

$4 x^{3} + 4 y^{3} y'$

Step 3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 4

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

$4 x^{3} + 4 y^{3} y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $4 x^{3}$ from both sides of the equation.

$4 y^{3} y' = - 4 x^{3}$

Step 5.2

Divide each term in $4 y^{3} y' = - 4 x^{3}$ by $4 y^{3}$ and simplify.

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Step 5.2.1

Divide each term in $4 y^{3} y' = - 4 x^{3}$ by $4 y^{3}$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of $y^{3}$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

Factor $4$ out of $- 4 x^{3}$ .

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

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $4$ out of $4 y^{3}$ .

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.2

Move the negative in front of the fraction.

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

Step 6

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

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