Calculus Examples

$y = {(\frac{1 + x^{2}}{1 - x^{2}})}^{17}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(\frac{1 + x^{2}}{1 - x^{2}})}^{17})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{17}$ and $g (x) = \frac{1 + x^{2}}{1 - x^{2}}$ .

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Step 3.1.1

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

$\frac{d}{d u} [u^{17}] \frac{d}{d x} [\frac{1 + x^{2}}{1 - x^{2}}]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 17$ .

$17 u^{16} \frac{d}{d x} [\frac{1 + x^{2}}{1 - x^{2}}]$

Step 3.1.3

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{d}{d x} [\frac{1 + x^{2}}{1 - x^{2}}]$

Step 3.2

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{(1 - x^{2}) \frac{d}{d x} [1 + x^{2}] - (1 + x^{2}) \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{(1 - x^{2}) (\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]) - (1 + x^{2}) \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.2

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{(1 - x^{2}) (0 + \frac{d}{d x} [x^{2}]) - (1 + x^{2}) \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.3

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{(1 - x^{2}) \frac{d}{d x} [x^{2}] - (1 + x^{2}) \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.4

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{(1 - x^{2}) (2 x) - (1 + x^{2}) \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.5

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 \cdot (1 - x^{2}) x - (1 + x^{2}) \frac{d}{d x} [1 - x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.6

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x - (1 + x^{2}) (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{2}}$

Step 3.3.7

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x - (1 + x^{2}) (0 + \frac{d}{d x} [- x^{2}])}{{(1 - x^{2})}^{2}}$

Step 3.3.8

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x - (1 + x^{2}) \frac{d}{d x} [- x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x - (1 + x^{2}) (- \frac{d}{d x} [x^{2}])}{{(1 - x^{2})}^{2}}$

Step 3.3.10

Multiply.

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Step 3.3.10.1

Multiply $- 1$ by $- 1$ .

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x + 1 (1 + x^{2}) \frac{d}{d x} [x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.10.2

Multiply $1 + x^{2}$ by $1$ .

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x + (1 + x^{2}) \frac{d}{d x} [x^{2}]}{{(1 - x^{2})}^{2}}$

Step 3.3.11

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x + (1 + x^{2}) (2 x)}{{(1 - x^{2})}^{2}}$

Step 3.3.12

Combine fractions.

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Step 3.3.12.1

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

$17 {(\frac{1 + x^{2}}{1 - x^{2}})}^{16} \frac{2 (1 - x^{2}) x + 2 \cdot (1 + x^{2}) x}{{(1 - x^{2})}^{2}}$

Step 3.3.12.2

Combine $\frac{2 (1 - x^{2}) x + 2 (1 + x^{2}) x}{{(1 - x^{2})}^{2}}$ and $17$ .

$\frac{(2 (1 - x^{2}) x + 2 (1 + x^{2}) x) \cdot 17}{{(1 - x^{2})}^{2}} {(\frac{1 + x^{2}}{1 - x^{2}})}^{16}$

Step 3.3.12.3

Move $17$ to the left of $2 (1 - x^{2}) x + 2 (1 + x^{2}) x$ .

$\frac{17 \cdot (2 (1 - x^{2}) x + 2 (1 + x^{2}) x)}{{(1 - x^{2})}^{2}} {(\frac{1 + x^{2}}{1 - x^{2}})}^{16}$

Step 3.4

Simplify.

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Step 3.4.1

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

$\frac{17 (2 (1 - x^{2}) x + 2 (1 + x^{2}) x)}{{(1 - x^{2})}^{2}} \cdot \frac{{(1 + x^{2})}^{16}}{{(1 - x^{2})}^{16}}$

Step 3.4.2