Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $2 x$ and $0$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

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

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

Step 8

Multiply $- 1$ by $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

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