Calculus Examples

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

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply.

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Step 2.10.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.2

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $2$ by $1$ .

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

Step 3.5.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.5.1.2.1

Move $x$ .

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

Step 3.5.1.2.2

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

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Step 3.5.1.2.2.1

Raise $x$ to the power of $1$ .

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

Step 3.5.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.1.2.3

Add $1$ and $2$ .

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

Step 3.5.1.3

Multiply $- 1$ by $2$ .

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

Step 3.5.1.4

Multiply $2$ by $1$ .

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

Step 3.5.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.5.1.5.1

Move $x$ .

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

Step 3.5.1.5.2

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

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Step 3.5.1.5.2.1

Raise $x$ to the power of $1$ .

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

Step 3.5.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.1.5.3

Add $1$ and $2$ .

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

Step 3.5.2

Combine the opposite terms in $2 x - 2 x^{3} + 2 x + 2 x^{3}$ .

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Step 3.5.2.1

Add $- 2 x^{3}$ and $2 x^{3}$ .

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

Step 3.5.2.2

Add $2 x + 2 x$ and $0$ .

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

Step 3.5.3

Add $2 x$ and $2 x$ .

$\frac{4 x}{{(1 - x^{2})}^{2}}$

Step 3.6

Reorder terms.

$\frac{4 x}{{(- x^{2} + 1)}^{2}}$

Step 3.7

Simplify the denominator.

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Step 3.7.1

Rewrite $1$ as $1^{2}$ .

$\frac{4 x}{{(- x^{2} + 1^{2})}^{2}}$

Step 3.7.2

Reorder $- x^{2}$ and $1^{2}$ .

$\frac{4 x}{{(1^{2} - x^{2})}^{2}}$

Step 3.7.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\frac{4 x}{{((1 + x) (1 - x))}^{2}}$

Step 3.7.4

Apply the product rule to $(1 + x) (1 - x)$ .

$\frac{4 x}{{(1 + x)}^{2} {(1 - x)}^{2}}$