Calculus Examples

$y = \cos (\frac{x - 1}{x + 1})$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{x - 1}{x + 1}]$

Step 1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$- \sin (u) \frac{d}{d x} [\frac{x - 1}{x + 1}]$

Step 1.3

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

$- \sin (\frac{x - 1}{x + 1}) \frac{d}{d x} [\frac{x - 1}{x + 1}]$

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 $f (x) = x - 1$ and $g (x) = x + 1$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $x + 1$ by $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Combine fractions.

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Step 3.8.1

Add $1$ and $0$ .

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

Step 3.8.2

Multiply $- 1$ by $1$ .

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

Step 3.8.3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Combine the opposite terms in $x + 1 - x - - 1$ .

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Step 4.2.1.1

Subtract $x$ from $x$ .

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

Step 4.2.1.2

Add $0$ and $1$ .

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

Step 4.2.2

Multiply $- 1$ by $- 1$ .

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

Step 4.2.3

Add $1$ and $1$ .

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