Calculus Examples

$f (x) = \cos^{3} (\frac{x}{x + 1})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (\frac{x}{x + 1})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (\frac{x}{x + 1})$ .

$\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\cos (\frac{x}{x + 1})]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$3 {u_{1}}^{2} \frac{d}{d x} [\cos (\frac{x}{x + 1})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cos (\frac{x}{x + 1})$ .

$3 \cos^{2} (\frac{x}{x + 1}) \frac{d}{d x} [\cos (\frac{x}{x + 1})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{x}{x + 1}$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{x + 1}$ .

$3 \cos^{2} (\frac{x}{x + 1}) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [\frac{x}{x + 1}])$

Step 2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$3 \cos^{2} (\frac{x}{x + 1}) (- \sin (u_{2}) \frac{d}{d x} [\frac{x}{x + 1}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\frac{x}{x + 1}$ .

$3 \cos^{2} (\frac{x}{x + 1}) (- \sin (\frac{x}{x + 1}) \frac{d}{d x} [\frac{x}{x + 1}])$

Step 3

Multiply $- 1$ by $3$ .

$- 3 \cos^{2} (\frac{x}{x + 1}) (\sin (\frac{x}{x + 1}) \frac{d}{d x} [\frac{x}{x + 1}])$

Step 4

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 $g (x) = x + 1$ .

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{(x + 1) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 5

Differentiate.

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Step 5.1

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

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{(x + 1) \cdot 1 - x \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 5.2

Multiply $x + 1$ by $1$ .

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{x + 1 - x \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 5.3

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

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{x + 1 - x (\frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 5.4

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

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{x + 1 - x (1 + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 5.5

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

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{x + 1 - x (1 + 0)}{{(x + 1)}^{2}}$

Step 5.6

Simplify terms.

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Step 5.6.1

Add $1$ and $0$ .

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{x + 1 - x \cdot 1}{{(x + 1)}^{2}}$

Step 5.6.2

Multiply $- 1$ by $1$ .

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{x + 1 - x}{{(x + 1)}^{2}}$

Step 5.6.3

Subtract $x$ from $x$ .

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{0 + 1}{{(x + 1)}^{2}}$

Step 5.6.4

Add $0$ and $1$ .

$- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1}) \frac{1}{{(x + 1)}^{2}}$

Step 5.6.5

Combine $\frac{1}{{(x + 1)}^{2}}$ and $- 3$ .

$\frac{- 3}{{(x + 1)}^{2}} \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1})$

Step 5.6.6

Combine $\frac{- 3}{{(x + 1)}^{2}}$ and $\cos^{2} (\frac{x}{x + 1})$ .

$\frac{- 3 \cos^{2} (\frac{x}{x + 1})}{{(x + 1)}^{2}} \sin (\frac{x}{x + 1})$

Step 5.6.7

Combine $\frac{- 3 \cos^{2} (\frac{x}{x + 1})}{{(x + 1)}^{2}}$ and $\sin (\frac{x}{x + 1})$ .

$\frac{- 3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1})}{{(x + 1)}^{2}}$

Step 5.6.8

Move the negative in front of the fraction.

$- \frac{3 \cos^{2} (\frac{x}{x + 1}) \sin (\frac{x}{x + 1})}{{(x + 1)}^{2}}$