Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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

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

Step 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 $n = 1$ .

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

Step 3.2

Multiply $x - 1$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Simplify terms.

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Step 3.6.1

Add $1$ and $0$ .

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

Step 3.6.2

Multiply $- 1$ by $1$ .

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

Step 3.6.3

Subtract $x$ from $x$ .

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

Step 3.6.4

Simplify the expression.

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Step 3.6.4.1

Subtract $1$ from $0$ .

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

Step 3.6.4.2

Move the negative in front of the fraction.

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

Step 3.6.4.3

Multiply $- 1$ by $- 1$ .

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

Step 3.6.4.4

Multiply $\sin (\frac{x}{x - 1})$ by $1$ .

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

Step 3.6.5

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

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