Calculus Examples

$\frac{x}{x^{2} - 8}$

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

$g (x) = x^{2} - 8$ .

$\frac{(x^{2} - 8) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 8]}{{(x^{2} - 8)}^{2}}$

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

$n = 1$ .

$\frac{(x^{2} - 8) \cdot 1 - x \frac{d}{d x} [x^{2} - 8]}{{(x^{2} - 8)}^{2}}$

Step 2.2

Multiply

$x^{2} - 8$ by

$1$ .

$\frac{x^{2} - 8 - x \frac{d}{d x} [x^{2} - 8]}{{(x^{2} - 8)}^{2}}$

Step 2.3

By the Sum Rule, the derivative of

$x^{2} - 8$ with respect to

$x$ is

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8]$ .

$\frac{x^{2} - 8 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8])}{{(x^{2} - 8)}^{2}}$

Step 2.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$\frac{x^{2} - 8 - x (2 x + \frac{d}{d x} [- 8])}{{(x^{2} - 8)}^{2}}$

Step 2.5

Since

$- 8$ is constant with respect to

$x$ , the derivative of

$- 8$ with respect to

$x$ is

$0$ .

$\frac{x^{2} - 8 - x (2 x + 0)}{{(x^{2} - 8)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add

$2 x$ and

$0$ .

$\frac{x^{2} - 8 - x (2 x)}{{(x^{2} - 8)}^{2}}$

Step 2.6.2

Multiply

$2$ by

$- 1$ .

$\frac{x^{2} - 8 - 2 x \cdot x}{{(x^{2} - 8)}^{2}}$

Step 3

Raise

$x$ to the power of

$1$ .

$\frac{x^{2} - 8 - 2 (x^{1} x)}{{(x^{2} - 8)}^{2}}$

Step 4

Raise

$x$ to the power of

$1$ .

$\frac{x^{2} - 8 - 2 (x^{1} x^{1})}{{(x^{2} - 8)}^{2}}$

Step 5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{2} - 8 - 2 x^{1 + 1}}{{(x^{2} - 8)}^{2}}$

Step 6

Add

$1$ and

$1$ .

$\frac{x^{2} - 8 - 2 x^{2}}{{(x^{2} - 8)}^{2}}$

Step 7

Subtract

$2 x^{2}$ from

$x^{2}$ .

$\frac{- x^{2} - 8}{{(x^{2} - 8)}^{2}}$

Step 8

Simplify.

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Step 8.1

Factor

$- 1$ out of

$- x^{2}$ .

$\frac{- (x^{2}) - 8}{{(x^{2} - 8)}^{2}}$

Step 8.2

Rewrite

$- 8$ as

$- 1 (8)$ .

$\frac{- (x^{2}) - 1 (8)}{{(x^{2} - 8)}^{2}}$

Step 8.3

Factor

$- 1$ out of

$- (x^{2}) - 1 (8)$ .

$\frac{- (x^{2} + 8)}{{(x^{2} - 8)}^{2}}$

Step 8.4

Rewrite

$- (x^{2} + 8)$ as

$- 1 (x^{2} + 8)$ .

$\frac{- 1 (x^{2} + 8)}{{(x^{2} - 8)}^{2}}$

Step 8.5

Move the negative in front of the fraction.

$- \frac{x^{2} + 8}{{(x^{2} - 8)}^{2}}$

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