Calculus Examples

$y = \frac{8}{x} + 3 \sec (x)$

Step 1

By the Sum Rule, the derivative of $\frac{8}{x} + 3 \sec (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{8}{x}] + \frac{d}{d x} [3 \sec (x)]$ .

$\frac{d}{d x} [\frac{8}{x}] + \frac{d}{d x} [3 \sec (x)]$

Step 2

Evaluate $\frac{d}{d x} [\frac{8}{x}]$ .

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x}]$ .

$8 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [3 \sec (x)]$

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$8 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [3 \sec (x)]$

Step 2.3

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

$8 (- x^{- 2}) + \frac{d}{d x} [3 \sec (x)]$

Step 2.4

Multiply $- 1$ by $8$ .

$- 8 x^{- 2} + \frac{d}{d x} [3 \sec (x)]$

Step 3

Evaluate $\frac{d}{d x} [3 \sec (x)]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \sec (x)$ with respect to $x$ is $3 \frac{d}{d x} [\sec (x)]$ .

$- 8 x^{- 2} + 3 \frac{d}{d x} [\sec (x)]$

Step 3.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$- 8 x^{- 2} + 3 \sec (x) \tan (x)$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 8 \frac{1}{x^{2}} + 3 \sec (x) \tan (x)$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 8$ and $\frac{1}{x^{2}}$ .

$\frac{- 8}{x^{2}} + 3 \sec (x) \tan (x)$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{8}{x^{2}} + 3 \sec (x) \tan (x)$

Step 4.3

Reorder terms.

$3 \sec (x) \tan (x) - \frac{8}{x^{2}}$