Calculus Examples

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

Step 1

Subtract $1$ from $2$ .

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

Step 2

Simplify.

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

Step 3

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 4

Differentiate.

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Step 4.1

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 4.2

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 4.3

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 4.4

Add $2 x$ and $0$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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 8

Add $1$ and $1$ .

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

Step 9

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 10

Multiply $- 1$ by $1$ .

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Simplify the numerator.

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Step 11.2.1

Multiply $- 1$ by $1$ .

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

Step 11.2.2

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

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

Step 11.3

Simplify the numerator.

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Step 11.3.1

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

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

Step 11.3.2

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

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