Calculus Examples

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

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

Step 1

Differentiate both sides of the equation.

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

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of

x^{2} + y^{2}

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

y

$y$ is

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

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

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

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

Step 2.2

Evaluate

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

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

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Step 2.2.1

Differentiate using the chain rule, which states that

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

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

$f' (g (y)) g' (y)$ where

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

$g (y) = x$ .

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Step 2.2.1.1

To apply the Chain Rule, set

$u$ as

$x$ .

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

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

$n = 2$ .

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

Step 2.2.1.3

Replace all occurrences of

$u$ with

$x$ .

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

Step 2.2.2

Rewrite

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

$x'$ .

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

Step 2.3

Differentiate using the Power Rule which states that

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

$n y^{n - 1}$ where

$n = 2$ .

$2 x x' + 2 y$

Step 3

Since

$25$ is constant with respect to

$y$ , the derivative of

$25$ with respect to

$y$ is

$0$ .

$0$

Step 4

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

$2 x x' + 2 y = 0$

Step 5

Solve for

$x'$ .

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Step 5.1

Subtract

$2 y$ from both sides of the equation.

$2 x x' = - 2 y$

Step 5.2

Divide each term in

$2 x x' = - 2 y$ by

$2 x$ and simplify.

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Step 5.2.1

Divide each term in

$2 x x' = - 2 y$ by

$2 x$ .

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

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.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of

$x$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide

$x'$ by

$1$ .

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

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

$- 2 y$ .

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

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

$2 x$ .

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.2

Move the negative in front of the fraction.

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

Step 6

Replace

$x'$ with

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

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

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