Calculus Examples

$f (x) = 8 \cos (x) - 3 \sec (x)$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 2.3

Multiply $- 1$ by $8$ .

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

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

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

Step 3.2

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

$- 8 \sin (x) - 3 \sec (x) \tan (x)$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$- 3 \sec (x) \tan (x) - 8 \sin (x)$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- 3 \frac{1}{\cos (x)} \tan (x) - 8 \sin (x)$

Step 4.2.2

Combine $- 3$ and $\frac{1}{\cos (x)}$ .

$\frac{- 3}{\cos (x)} \tan (x) - 8 \sin (x)$

Step 4.2.3

Move the negative in front of the fraction.

$- \frac{3}{\cos (x)} \tan (x) - 8 \sin (x)$

Step 4.2.4

Rewrite $\tan (x)$ in terms of sines and cosines.

$- \frac{3}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} - 8 \sin (x)$

Step 4.2.5

Multiply $- \frac{3}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 4.2.5.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{3}{\cos (x)}$ .

$- \frac{\sin (x) \cdot 3}{\cos (x) \cos (x)} - 8 \sin (x)$

Step 4.2.5.2

Raise $\cos (x)$ to the power of $1$ .

$- \frac{\sin (x) \cdot 3}{\cos^{1} (x) \cos (x)} - 8 \sin (x)$

Step 4.2.5.3

Raise $\cos (x)$ to the power of $1$ .

$- \frac{\sin (x) \cdot 3}{\cos^{1} (x) \cos^{1} (x)} - 8 \sin (x)$

Step 4.2.5.4

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

$- \frac{\sin (x) \cdot 3}{{\cos (x)}^{1 + 1}} - 8 \sin (x)$

Step 4.2.5.5

Add $1$ and $1$ .

$- \frac{\sin (x) \cdot 3}{\cos^{2} (x)} - 8 \sin (x)$

Step 4.2.6

Move $3$ to the left of $\sin (x)$ .

$- \frac{3 \sin (x)}{\cos^{2} (x)} - 8 \sin (x)$

Step 4.3

Simplify each term.

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Step 4.3.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$- \frac{3 \sin (x)}{\cos (x) \cos (x)} - 8 \sin (x)$

Step 4.3.2

Separate fractions.

$- (\frac{3}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}) - 8 \sin (x)$

Step 4.3.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$- (\frac{3}{\cos (x)} \tan (x)) - 8 \sin (x)$

Step 4.3.4

Separate fractions.

$- (\frac{3}{1} \cdot \frac{1}{\cos (x)} \tan (x)) - 8 \sin (x)$

Step 4.3.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- (\frac{3}{1} \sec (x) \tan (x)) - 8 \sin (x)$

Step 4.3.6

Divide $3$ by $1$ .

$- (3 \sec (x) \tan (x)) - 8 \sin (x)$

Step 4.3.7

Multiply $3$ by $- 1$ .

$- 3 \sec (x) \tan (x) - 8 \sin (x)$