Calculus Examples

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

Step 1

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 $g (x) = \frac{x}{1 + x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x}{1 + x}$ .

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [\frac{x}{1 + x}]$

Step 1.2

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

$5 u^{4} \frac{d}{d x} [\frac{x}{1 + x}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x}{1 + x}$ .

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

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 $g (x) = 1 + x$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{(1 + x) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + x]}{{(1 + x)}^{2}}$

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 $n = 1$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{(1 + x) \cdot 1 - x \frac{d}{d x} [1 + x]}{{(1 + x)}^{2}}$

Step 3.2

Multiply $1 + x$ by $1$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{1 + x - x \frac{d}{d x} [1 + x]}{{(1 + x)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $1 + x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x]$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{1 + x - x (\frac{d}{d x} [1] + \frac{d}{d x} [x])}{{(1 + x)}^{2}}$

Step 3.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{1 + x - x (0 + \frac{d}{d x} [x])}{{(1 + x)}^{2}}$

Step 3.5

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

$5 {(\frac{x}{1 + x})}^{4} \frac{1 + x - x \frac{d}{d x} [x]}{{(1 + x)}^{2}}$

Step 3.6

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

$5 {(\frac{x}{1 + x})}^{4} \frac{1 + x - x \cdot 1}{{(1 + x)}^{2}}$

Step 3.7

Simplify terms.

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Step 3.7.1

Multiply $- 1$ by $1$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{1 + x - x}{{(1 + x)}^{2}}$

Step 3.7.2

Subtract $x$ from $x$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{1 + 0}{{(1 + x)}^{2}}$

Step 3.7.3

Add $1$ and $0$ .

$5 {(\frac{x}{1 + x})}^{4} \frac{1}{{(1 + x)}^{2}}$

Step 3.7.4

Combine $\frac{1}{{(1 + x)}^{2}}$ and $5$ .

$\frac{5}{{(1 + x)}^{2}} {(\frac{x}{1 + x})}^{4}$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{x}{1 + x}$ .

$\frac{5}{{(1 + x)}^{2}} \cdot \frac{x^{4}}{{(1 + x)}^{4}}$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $\frac{5}{{(1 + x)}^{2}}$ by $\frac{x^{4}}{{(1 + x)}^{4}}$ .

$\frac{5 x^{4}}{{(1 + x)}^{2} {(1 + x)}^{4}}$

Step 4.2.2

Multiply ${(1 + x)}^{2}$ by ${(1 + x)}^{4}$ by adding the exponents.

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Step 4.2.2.1

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

$\frac{5 x^{4}}{{(1 + x)}^{2 + 4}}$

Step 4.2.2.2

Add $2$ and $4$ .

$\frac{5 x^{4}}{{(1 + x)}^{6}}$

Step 4.3

Reorder terms.

$\frac{5 x^{4}}{{(x + 1)}^{6}}$