Calculus Examples

$\frac{- y}{x^{2} + y^{2}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

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

Step 1.2

Since $- y$ is constant with respect to $x$ , the derivative of $- \frac{y}{x^{2} + y^{2}}$ with respect to $x$ is $- y \frac{d}{d x} [\frac{1}{x^{2} + y^{2}}]$ .

$- y \frac{d}{d x} [\frac{1}{x^{2} + y^{2}}]$

Step 1.3

Rewrite $\frac{1}{x^{2} + y^{2}}$ as ${(x^{2} + y^{2})}^{- 1}$ .

$- y \frac{d}{d x} [{(x^{2} + y^{2})}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} + y^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2} + y^{2}$ .

$- y (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x^{2} + y^{2}$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $- 1$ .

$1 y ({(x^{2} + y^{2})}^{- 2} \frac{d}{d x} [x^{2} + y^{2}])$

Step 3.2

Multiply $y$ by $1$ .

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

Step 3.3

By the Sum Rule, the derivative of $x^{2} + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$ .

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

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.5

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$y ({(x^{2} + y^{2})}^{- 2} (2 x + 0))$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2 x$ and $0$ .

$y ({(x^{2} + y^{2})}^{- 2} (2 x))$

Step 3.6.2

Move $2$ to the left of ${(x^{2} + y^{2})}^{- 2}$ .

$y (2 \cdot {(x^{2} + y^{2})}^{- 2} x)$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y (2 \frac{1}{{(x^{2} + y^{2})}^{2}} x)$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $2$ and $\frac{1}{{(x^{2} + y^{2})}^{2}}$ .

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

Step 4.2.2

Combine $\frac{2}{{(x^{2} + y^{2})}^{2}}$ and $x$ .

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

Step 4.2.3

Combine $y$ and $\frac{2 x}{{(x^{2} + y^{2})}^{2}}$ .

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

Step 4.2.4

Move $2$ to the left of $y$ .

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

Step 4.3

Reorder terms.

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