Calculus Examples

$f (x) = - \frac{4 h}{x (x + h)}$

Step 1

Since $- \frac{4}{x}$ is constant with respect to $h$ , the derivative of $- \frac{4 h}{x (x + h)}$ with respect to $h$ is $- \frac{4}{x} \frac{d}{d h} [\frac{h}{x + h}]$ .

$- \frac{4}{x} \frac{d}{d h} [\frac{h}{x + h}]$

Step 2

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

$- \frac{4}{x} \cdot \frac{(x + h) \frac{d}{d h} [h] - h \frac{d}{d h} [x + h]}{{(x + h)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$- \frac{4}{x} \cdot \frac{(x + h) \cdot 1 - h \frac{d}{d h} [x + h]}{{(x + h)}^{2}}$

Step 3.2

Multiply $x + h$ by $1$ .

$- \frac{4}{x} \cdot \frac{x + h - h \frac{d}{d h} [x + h]}{{(x + h)}^{2}}$

Step 3.3

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

$- \frac{4}{x} \cdot \frac{x + h - h (\frac{d}{d h} [x] + \frac{d}{d h} [h])}{{(x + h)}^{2}}$

Step 3.4

Since $x$ is constant with respect to $h$ , the derivative of $x$ with respect to $h$ is $0$ .

$- \frac{4}{x} \cdot \frac{x + h - h (0 + \frac{d}{d h} [h])}{{(x + h)}^{2}}$

Step 3.5

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

$- \frac{4}{x} \cdot \frac{x + h - h \frac{d}{d h} [h]}{{(x + h)}^{2}}$

Step 3.6

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

$- \frac{4}{x} \cdot \frac{x + h - h \cdot 1}{{(x + h)}^{2}}$

Step 3.7

Simplify terms.

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Step 3.7.1

Multiply $- 1$ by $1$ .

$- \frac{4}{x} \cdot \frac{x + h - h}{{(x + h)}^{2}}$

Step 3.7.2

Subtract $h$ from $h$ .

$- \frac{4}{x} \cdot \frac{x + 0}{{(x + h)}^{2}}$

Step 3.7.3

Add $x$ and $0$ .

$- \frac{4}{x} \cdot \frac{x}{{(x + h)}^{2}}$

Step 3.7.4

Multiply $\frac{x}{{(x + h)}^{2}}$ by $\frac{4}{x}$ .

$- \frac{x \cdot 4}{{(x + h)}^{2} x}$

Step 3.7.5

Move $4$ to the left of $x$ .

$- \frac{4 \cdot x}{{(x + h)}^{2} x}$

Step 3.7.6

Cancel the common factor of $x$ .

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Step 3.7.6.1

Cancel the common factor.

$- \frac{4 x}{{(x + h)}^{2} x}$

Step 3.7.6.2

Rewrite the expression.

$- \frac{4}{{(x + h)}^{2}}$