Calculus Examples

x^{3} + y^{3} = 4

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

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (x^{3} + y^{3}) = \frac{d}{d x} (4)

$\frac{d}{d x} (x^{3} + y^{3}) = \frac{d}{d x} (4)$

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^{3} + y^{3}

$x^{3} + y^{3}$ with respect to

x

$x$ is

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

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

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

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

Step 2.1.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

3 x^{2} + \frac{d}{d x} [y^{3}]

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

3 x^{2} + \frac{d}{d x} [y^{3}]

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

Step 2.2

Evaluate

\frac{d}{d x} [y^{3}]

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

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Step 2.2.1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{3}

$f (x) = x^{3}$ and

g (x) = y

$g (x) = y$ .

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Step 2.2.1.1

To apply the Chain Rule, set

u

$u$ as

y

$y$ .

3 x^{2} + \frac{d}{d u} [u^{3}] \frac{d}{d x} [y]

$3 x^{2} + \frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

Step 2.2.1.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = 3

$n = 3$ .

3 x^{2} + 3 u^{2} \frac{d}{d x} [y]

$3 x^{2} + 3 u^{2} \frac{d}{d x} [y]$

Step 2.2.1.3

Replace all occurrences of

u

$u$ with

y

$y$ .

3 x^{2} + 3 y^{2} \frac{d}{d x} [y]

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

3 x^{2} + 3 y^{2} \frac{d}{d x} [y]

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

Step 2.2.2

Rewrite

\frac{d}{d x} [y]

$\frac{d}{d x} [y]$ as

y'

$y'$ .

3 x^{2} + 3 y^{2} y'

$3 x^{2} + 3 y^{2} y'$

3 x^{2} + 3 y^{2} y'

$3 x^{2} + 3 y^{2} y'$

3 x^{2} + 3 y^{2} y'

$3 x^{2} + 3 y^{2} y'$

Step 3

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4

$4$ with respect to

x

$x$ is

0

$0$ .

0

$0$

Step 4

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

3 x^{2} + 3 y^{2} y' = 0

$3 x^{2} + 3 y^{2} y' = 0$

Step 5

Solve for

y'

$y'$ .

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Step 5.1

Subtract

3 x^{2}

$3 x^{2}$ from both sides of the equation.

3 y^{2} y' = - 3 x^{2}

$3 y^{2} y' = - 3 x^{2}$

Step 5.2

Divide each term in

3 y^{2} y' = - 3 x^{2}

$3 y^{2} y' = - 3 x^{2}$ by

3 y^{2}

$3 y^{2}$ and simplify.

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Step 5.2.1

Divide each term in

3 y^{2} y' = - 3 x^{2}

$3 y^{2} y' = - 3 x^{2}$ by

3 y^{2}

$3 y^{2}$ .

\frac{3 y^{2} y'}{3 y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}

$\frac{3 y^{2} y'}{3 y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

3

$3$ .

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Step 5.2.2.1.1

Cancel the common factor.

\frac{3 y^{2} y'}{3 y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}

$\frac{3 y^{2} y'}{3 y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}$

Step 5.2.2.1.2

Rewrite the expression.

\frac{y^{2} y'}{y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}

$\frac{y^{2} y'}{y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}$

\frac{y^{2} y'}{y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}

$\frac{y^{2} y'}{y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}$

Step 5.2.2.2

Cancel the common factor of

y^{2}

$y^{2}$ .

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Step 5.2.2.2.1

Cancel the common factor.

\frac{y^{2} y'}{y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}

$\frac{y^{2} y'}{y^{2}} = \frac{- 3 x^{2}}{3 y^{2}}$

Step 5.2.2.2.2

Divide

y'

$y'$ by

1

$1$ .

y' = \frac{- 3 x^{2}}{3 y^{2}}

$y' = \frac{- 3 x^{2}}{3 y^{2}}$

y' = \frac{- 3 x^{2}}{3 y^{2}}

$y' = \frac{- 3 x^{2}}{3 y^{2}}$

y' = \frac{- 3 x^{2}}{3 y^{2}}

$y' = \frac{- 3 x^{2}}{3 y^{2}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of

- 3

$- 3$ and

3

$3$ .

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Step 5.2.3.1.1

Factor

3

$3$ out of

- 3 x^{2}

$- 3 x^{2}$ .

y' = \frac{3 (- x^{2})}{3 y^{2}}

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

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor

3

$3$ out of

3 y^{2}

$3 y^{2}$ .

y' = \frac{3 (- x^{2})}{3 (y^{2})}

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

Step 5.2.3.1.2.2

Cancel the common factor.

y' = \frac{3 (- x^{2})}{3 y^{2}}

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

Step 5.2.3.1.2.3

Rewrite the expression.

y' = \frac{- x^{2}}{y^{2}}

$y' = \frac{- x^{2}}{y^{2}}$

y' = \frac{- x^{2}}{y^{2}}

$y' = \frac{- x^{2}}{y^{2}}$

y' = \frac{- x^{2}}{y^{2}}

$y' = \frac{- x^{2}}{y^{2}}$

Step 5.2.3.2

Move the negative in front of the fraction.

y' = - \frac{x^{2}}{y^{2}}

$y' = - \frac{x^{2}}{y^{2}}$

y' = - \frac{x^{2}}{y^{2}}

$y' = - \frac{x^{2}}{y^{2}}$

y' = - \frac{x^{2}}{y^{2}}

$y' = - \frac{x^{2}}{y^{2}}$

y' = - \frac{x^{2}}{y^{2}}

$y' = - \frac{x^{2}}{y^{2}}$

Step 6

Replace

y'

$y'$ with

\frac{d y}{d x}

$\frac{d y}{d x}$ .

\frac{d y}{d x} = - \frac{x^{2}}{y^{2}}

$\frac{d y}{d x} = - \frac{x^{2}}{y^{2}}$

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