Calculus Examples

$\frac{x}{1 - x^{2}}$

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply.

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

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

Step 2.7.2

Multiply $x$ by $1$ .

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

Step 2.8

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

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

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

Step 8

Reorder terms.

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