Calculus Examples

x^{3} - y^{3} = 7

$x^{3} - y^{3} = 7$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (x^{3} - y^{3}) = \frac{d}{d x} (7)

$\frac{d}{d x} (x^{3} - y^{3}) = \frac{d}{d x} (7)$

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

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- y^{3}

$- y^{3}$ with respect to

x

$x$ is

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

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

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

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

Step 2.2.2

Differentiate using the chain rule, which states that

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

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

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

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

$g (x) = y$ .

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Step 2.2.2.1

To apply the Chain Rule, set

$u$ as

$y$ .

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

Step 2.2.2.2

Differentiate using the Power Rule which states that

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

$n u^{n - 1}$ where

$n = 3$ .

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

Step 2.2.2.3

Replace all occurrences of

$u$ with

$y$ .

$3 x^{2} - (3 y^{2} \frac{d}{d x} [y])$

Step 2.2.3

Rewrite

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

$y'$ .

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

Step 2.2.4

Multiply

$3$ by

$- 1$ .

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

Step 3

Since

$7$ is constant with respect to

$x$ , the derivative of

$7$ with respect to

$x$ is

$0$ .

$0$

Step 4

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

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

Step 5

Solve for

$y'$ .

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Step 5.1

Subtract

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

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

Step 5.2

Divide each term in

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

$- 3 y^{2}$ and simplify.

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Step 5.2.1

Divide each term in

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

$- 3 y^{2}$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$- 3$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of

$y^{2}$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide

$y'$ by

$1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of

$- 3$ .

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Step 5.2.3.1.1

Cancel the common factor.

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

Step 5.2.3.1.2

Rewrite the expression.

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

Step 6

Replace

$y'$ with

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

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