Calculus Examples

$f (x) = \cos (\tan (x))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \tan (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\tan (x)$ .

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

Step 1.2

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

$- \sin (u) \frac{d}{d x} [\tan (x)]$

Step 1.3

Replace all occurrences of $u$ with $\tan (x)$ .

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

Step 2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 3

Simplify.

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Step 3.1

Reorder the factors of $- \sin (\tan (x)) \sec^{2} (x)$ .

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

Step 3.2

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

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

Step 3.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 3.4

One to any power is one.

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

Step 3.5

Combine $\sin (\tan (x))$ and $\frac{1}{\cos^{2} (x)}$ .

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

Step 3.6

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

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

Step 3.7

Separate fractions.

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

Step 3.8

Rewrite $\frac{\sin (\tan (x))}{\cos (x)}$ as a product.

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

Step 3.9

Write $\sin (\tan (x))$ as a fraction with denominator $1$ .

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

Step 3.10

Simplify.

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Step 3.10.1

Divide $\sin (\tan (x))$ by $1$ .

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

Step 3.10.2

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

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

Step 3.11

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

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

Step 3.12

Multiply $\sec (x) (\sin (\tan (x)) \sec (x))$ .

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Step 3.12.1

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

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

Step 3.12.2

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

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

Step 3.12.3

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

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

Step 3.12.4

Add $1$ and $1$ .

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

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