Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec (x)$ and $g (x) = \sin (x)$ .

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

Step 3

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

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

Step 4

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

$- 3 (\sec (x) \cos (x) + \sin (x) (\sec (x) \tan (x)))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Remove parentheses.

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

Step 5.3

Reorder terms.

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

Step 5.4

Simplify each term.

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Step 5.4.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 5.4.1.1

Add parentheses.

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

Step 5.4.1.2

Reorder $\cos (x)$ and $\sec (x)$ .

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

Step 5.4.1.3

Rewrite $- 3 \cos (x) \sec (x)$ in terms of sines and cosines.

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

Step 5.4.1.4

Cancel the common factors.

$- 3 \cdot 1 - 3 \sec (x) \sin (x) \tan (x)$

Step 5.4.2

Multiply $- 3$ by $1$ .

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

Step 5.4.3

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

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

Step 5.4.4

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

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

Step 5.4.5

Move the negative in front of the fraction.

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

Step 5.4.6

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

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

Step 5.4.7

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

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

Step 5.4.8

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

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

Step 5.4.9

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

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Step 5.4.9.1

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

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

Step 5.4.9.2

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

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

Step 5.4.9.3

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

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

Step 5.4.9.4

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

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

Step 5.4.9.5

Add $1$ and $1$ .

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

Step 5.4.9.6

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

$- 3 - \frac{3 \sin^{2} (x)}{\cos^{1} (x) \cos (x)}$

Step 5.4.9.7

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

$- 3 - \frac{3 \sin^{2} (x)}{\cos^{1} (x) \cos^{1} (x)}$

Step 5.4.9.8

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

$- 3 - \frac{3 \sin^{2} (x)}{{\cos (x)}^{1 + 1}}$

Step 5.4.9.9

Add $1$ and $1$ .

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

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply by $1$ .

$- 3 - \frac{3 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x)}$

Step 5.5.2

Multiply by $1$ .

$- 3 - \frac{3 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cdot 1}$

Step 5.5.3

Separate fractions.

$- 3 - (\frac{3 \cdot (1)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)})$

Step 5.5.4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

$- 3 - (\frac{3 \cdot (1)}{1} \tan^{2} (x))$

Step 5.5.5

Multiply $3$ by $1$ .

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

Step 5.5.6

Divide $3$ by $1$ .

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

Step 5.5.7

Multiply $3$ by $- 1$ .

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

Step 5.6

Factor $- 3$ out of $- 3$ .

$- 3 (1) - 3 \tan^{2} (x)$

Step 5.7

Factor $- 3$ out of $- 3 \tan^{2} (x)$ .

$- 3 (1) - 3 (\tan^{2} (x))$

Step 5.8

Factor $- 3$ out of $- 3 (1) - 3 (\tan^{2} (x))$ .

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

Step 5.9

Rearrange terms.

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

Step 5.10

Apply pythagorean identity.

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