Calculus Examples

$\frac{5 x y}{y^{2} + 2 x}$

Step 1

Since $5 y$ is constant with respect to $x$ , the derivative of $\frac{5 x y}{y^{2} + 2 x}$ with respect to $x$ is $5 y \frac{d}{d x} [\frac{x}{y^{2} + 2 x}]$ .

$5 y \frac{d}{d x} [\frac{x}{y^{2} + 2 x}]$

Step 2

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

$5 y \frac{(y^{2} + 2 x) \frac{d}{d x} [x] - x \frac{d}{d x} [y^{2} + 2 x]}{{(y^{2} + 2 x)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$5 y \frac{(y^{2} + 2 x) \cdot 1 - x \frac{d}{d x} [y^{2} + 2 x]}{{(y^{2} + 2 x)}^{2}}$

Step 3.2

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

$5 y \frac{y^{2} + 2 x - x \frac{d}{d x} [y^{2} + 2 x]}{{(y^{2} + 2 x)}^{2}}$

Step 3.3

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

$5 y \frac{y^{2} + 2 x - x (\frac{d}{d x} [y^{2}] + \frac{d}{d x} [2 x])}{{(y^{2} + 2 x)}^{2}}$

Step 3.4

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

$5 y \frac{y^{2} + 2 x - x (0 + \frac{d}{d x} [2 x])}{{(y^{2} + 2 x)}^{2}}$

Step 3.5

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

$5 y \frac{y^{2} + 2 x - x \frac{d}{d x} [2 x]}{{(y^{2} + 2 x)}^{2}}$

Step 3.6

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

$5 y \frac{y^{2} + 2 x - x (2 \frac{d}{d x} [x])}{{(y^{2} + 2 x)}^{2}}$

Step 3.7

Multiply $2$ by $- 1$ .

$5 y \frac{y^{2} + 2 x - 2 x \frac{d}{d x} [x]}{{(y^{2} + 2 x)}^{2}}$

Step 3.8

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

$5 y \frac{y^{2} + 2 x - 2 x \cdot 1}{{(y^{2} + 2 x)}^{2}}$

Step 3.9

Simplify terms.

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Step 3.9.1

Multiply $- 2$ by $1$ .

$5 y \frac{y^{2} + 2 x - 2 x}{{(y^{2} + 2 x)}^{2}}$

Step 3.9.2

Subtract $2 x$ from $2 x$ .

$5 y \frac{y^{2} + 0}{{(y^{2} + 2 x)}^{2}}$

Step 3.9.3

Add $y^{2}$ and $0$ .

$5 y \frac{y^{2}}{{(y^{2} + 2 x)}^{2}}$

Step 3.9.4

Combine $5$ and $\frac{y^{2}}{{(y^{2} + 2 x)}^{2}}$ .

$y \frac{5 y^{2}}{{(y^{2} + 2 x)}^{2}}$

Step 3.9.5

Combine $y$ and $\frac{5 y^{2}}{{(y^{2} + 2 x)}^{2}}$ .

$\frac{y (5 y^{2})}{{(y^{2} + 2 x)}^{2}}$

Step 4

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

$\frac{5 (y^{1} y^{2})}{{(y^{2} + 2 x)}^{2}}$

Step 5

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

$\frac{5 y^{1 + 2}}{{(y^{2} + 2 x)}^{2}}$

Step 6

Add $1$ and $2$ .

$\frac{5 y^{3}}{{(y^{2} + 2 x)}^{2}}$