Calculus Examples

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

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Multiply $y^{2} + x^{2}$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $2 y$ and $0$ .

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

Step 2.6.2

Multiply $2$ by $- 1$ .

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

Step 3

Raise $y$ to the power of $1$ .

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

Step 4

Raise $y$ to the power of $1$ .

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

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6

Add $1$ and $1$ .

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

Step 7

Subtract $2 y^{2}$ from $y^{2}$ .

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

Step 8

Reorder terms.

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