Calculus Examples

x^{2} + y^{2} = 25

$x^{2} + y^{2} = 25$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (x^{2} + y^{2}) = \frac{d}{d x} (25)

$\frac{d}{d x} (x^{2} + y^{2}) = \frac{d}{d x} (25)$

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^{2} + y^{2}

$x^{2} + y^{2}$ with respect to

x

$x$ is

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

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

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

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

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 = 2

$n = 2$ .

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

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

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

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

Step 2.2

Evaluate

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

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

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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

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

$f (x) = x^{2}$ and

$g (x) = y$ .

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Step 2.2.1.1

To apply the Chain Rule, set

$u$ as

$y$ .

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

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

$n = 2$ .

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

Step 2.2.1.3

Replace all occurrences of

$u$ with

$y$ .

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

Step 2.2.2

Rewrite

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

$y'$ .

$2 x + 2 y y'$

Step 2.3

Reorder terms.

$2 y y' + 2 x$

Step 3

Since

$25$ is constant with respect to

$x$ , the derivative of

$25$ with respect to

$x$ is

$0$ .

$0$

Step 4

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

$2 y y' + 2 x = 0$

Step 5

Solve for

$y'$ .

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Step 5.1

Subtract

$2 x$ from both sides of the equation.

$2 y y' = - 2 x$

Step 5.2

Divide each term in

$2 y y' = - 2 x$ by

$2 y$ and simplify.

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Step 5.2.1

Divide each term in

$2 y y' = - 2 x$ by

$2 y$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of

$y$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide

$y'$ by

$1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of

$- 2$ and

$2$ .

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Step 5.2.3.1.1

Factor

$2$ out of

$- 2 x$ .

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

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor

$2$ out of

$2 y$ .

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.2

Move the negative in front of the fraction.

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

Step 6

Replace

$y'$ with

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

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