Calculus Examples

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

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 $g (y) = x^{2} + y^{2}$ .

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

Differentiate.

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

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

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

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

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

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

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

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

Add $0$ and $\frac{d}{d y} [y^{2}]$ .

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

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

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

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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