Calculus Examples

$y = {(\frac{x^{2} + 1}{x^{2} - 1})}^{3}$

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 $g (x) = \frac{x^{2} + 1}{x^{2} - 1}$ .

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Step 1.1

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

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [\frac{x^{2} + 1}{x^{2} - 1}]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [\frac{x^{2} + 1}{x^{2} - 1}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x^{2} + 1}{x^{2} - 1}$ .

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

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 $f (x) = x^{2} + 1$ and $g (x) = x^{2} - 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.3

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

$3 {(\frac{x^{2} + 1}{x^{2} - 1})}^{2} \frac{(x^{2} - 1) (2 x + 0) - (x^{2} + 1) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $2 x$ and $0$ .

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

Step 3.4.2

Move $2$ to the left of $x^{2} - 1$ .

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

Step 3.5

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.7

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

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

Step 3.8

Combine fractions.

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Step 3.8.1

Add $2 x$ and $0$ .

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

Step 3.8.2

Multiply $2$ by $- 1$ .

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

Step 3.8.3

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

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

Step 3.8.4

Move $3$ to the left of $2 (x^{2} - 1) x - 2 (x^{2} + 1) x$ .

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

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{x^{2} + 1}{x^{2} - 1}$ .

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

Step 4.2