Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

Since $\frac{1}{h}$ is constant with respect to $x$ , the derivative of $\frac{{(x + h)}^{5} - x^{5}}{h}$ with respect to $x$ is $\frac{1}{h} \frac{d}{d x} [{(x + h)}^{5} - x^{5}]$ .

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

Step 1.2

By the Sum Rule, the derivative of ${(x + h)}^{5} - x^{5}$ with respect to $x$ is $\frac{d}{d x} [{(x + h)}^{5}] + \frac{d}{d x} [- x^{5}]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x + h$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x + h$ .

$\frac{1}{h} (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x + h] + \frac{d}{d x} [- x^{5}])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$\frac{1}{h} (5 u^{4} \frac{d}{d x} [x + h] + \frac{d}{d x} [- x^{5}])$

Step 2.3

Replace all occurrences of $u$ with $x + h$ .

$\frac{1}{h} (5 {(x + h)}^{4} \frac{d}{d x} [x + h] + \frac{d}{d x} [- x^{5}])$

Step 3

Differentiate.

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Step 3.1

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

$\frac{1}{h} (5 {(x + h)}^{4} (\frac{d}{d x} [x] + \frac{d}{d x} [h]) + \frac{d}{d x} [- x^{5}])$

Step 3.2

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

$\frac{1}{h} (5 {(x + h)}^{4} (1 + \frac{d}{d x} [h]) + \frac{d}{d x} [- x^{5}])$

Step 3.3

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

$\frac{1}{h} (5 {(x + h)}^{4} (1 + 0) + \frac{d}{d x} [- x^{5}])$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$\frac{1}{h} (5 {(x + h)}^{4} \cdot 1 + \frac{d}{d x} [- x^{5}])$

Step 3.4.2

Multiply $5$ by $1$ .

$\frac{1}{h} (5 {(x + h)}^{4} + \frac{d}{d x} [- x^{5}])$

Step 3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{5}$ with respect to $x$ is $- \frac{d}{d x} [x^{5}]$ .

$\frac{1}{h} (5 {(x + h)}^{4} - \frac{d}{d x} [x^{5}])$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{h} (5 {(x + h)}^{4} - (5 x^{4}))$

Step 3.7

Multiply $5$ by $- 1$ .

$\frac{1}{h} (5 {(x + h)}^{4} - 5 x^{4})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{1}{h} (5 {(x + h)}^{4}) + \frac{1}{h} (- 5 x^{4})$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $5$ and $\frac{1}{h}$ .

$\frac{5}{h} {(x + h)}^{4} + \frac{1}{h} (- 5 x^{4})$

Step 4.2.2

Combine $\frac{5}{h}$ and ${(x + h)}^{4}$ .

$\frac{5 {(x + h)}^{4}}{h} + \frac{1}{h} (- 5 x^{4})$

Step 4.2.3

Combine $- 5$ and $\frac{1}{h}$ .

$\frac{5 {(x + h)}^{4}}{h} + \frac{- 5}{h} x^{4}$

Step 4.2.4

Combine $\frac{- 5}{h}$ and $x^{4}$ .

$\frac{5 {(x + h)}^{4}}{h} + \frac{- 5 x^{4}}{h}$

Step 4.2.5

Move the negative in front of the fraction.

$\frac{5 {(x + h)}^{4}}{h} - \frac{5 x^{4}}{h}$

Step 4.2.6

Combine the numerators over the common denominator.

$\frac{5 {(x + h)}^{4} - 5 x^{4}}{h}$

Step 4.3

Reorder terms.

$\frac{- 5 x^{4} + 5 {(x + h)}^{4}}{h}$