Calculus Examples

$f (x) = (\sin (x) + \cos (x)) \sec (x)$

Step 1

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 $f (x) = \sin (x) + \cos (x)$ and $g (x) = \sec (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Reorder terms.

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

Step 6.5

Simplify each term.

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Step 6.5.1

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

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

Step 6.5.2

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

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

Step 6.5.3

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

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

Step 6.5.4

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

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Step 6.5.4.1

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

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

Step 6.5.4.2

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

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

Step 6.5.4.3

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

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

Step 6.5.4.4

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

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

Step 6.5.4.5

Add $1$ and $1$ .

$\frac{\sin^{2} (x)}{\cos (x) \cos (x)} + \cos (x) \sec (x) \tan (x) + \cos (x) \sec (x) - \sec (x) \sin (x)$

Step 6.5.4.6

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

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

Step 6.5.4.7

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

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

Step 6.5.4.8

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

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

Step 6.5.4.9

Add $1$ and $1$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + \cos (x) \sec (x) \tan (x) + \cos (x) \sec (x) - \sec (x) \sin (x)$

Step 6.5.5

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

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Step 6.5.5.1

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + \sec (x) \cos (x) \tan (x) + \cos (x) \sec (x) - \sec (x) \sin (x)$

Step 6.5.5.2

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

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

Step 6.5.5.3

Cancel the common factors.

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

Step 6.5.6

Multiply $\tan (x)$ by $1$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan (x) + \cos (x) \sec (x) - \sec (x) \sin (x)$

Step 6.5.7

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

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

Step 6.5.8

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

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Step 6.5.8.1

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

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

Step 6.5.8.2

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

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

Step 6.5.8.3

Cancel the common factors.

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

Step 6.5.9

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

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

Step 6.5.10

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

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

Step 6.6

Combine the opposite terms in $\frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} + 1 - \frac{\sin (x)}{\cos (x)}$ .

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Step 6.6.1

Subtract $\frac{\sin (x)}{\cos (x)}$ from $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + 0 + 1$

Step 6.6.2

Add $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ and $0$ .

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

Step 6.7

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

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

Step 6.8

Apply pythagorean identity.

$\sec^{2} (x)$