Calculus Examples

$\cos (x) \sec (x)$

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Cancel the common factors.

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

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

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

Step 4.3

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

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