Calculus Examples

$\sin (x) \sec (x)$

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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Step 4.2.4.1

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

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

Step 4.2.4.2

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

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

Step 4.2.4.3

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

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

Step 4.2.4.4

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

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

Step 4.2.4.5

Add $1$ and $1$ .

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

Step 4.2.4.6

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

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

Step 4.2.4.7

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

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

Step 4.2.4.8

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

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

Step 4.2.4.9

Add $1$ and $1$ .

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

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.5.3

Cancel the common factors.

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

Step 4.3

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

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

Step 4.4

Apply pythagorean identity.

$\sec^{2} (x)$