Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (\tan (x))$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [\cos (\tan (x))]$

Step 1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [\cos (\tan (x))]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cos (\tan (x))$ .

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

Step 2

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 2.1

To apply the Chain Rule, set $u_{2}$ as $\tan (x)$ .

$\cos (\cos (\tan (x))) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [\tan (x)])$

Step 2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$\cos (\cos (\tan (x))) (- \sin (u_{2}) \frac{d}{d x} [\tan (x)])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\tan (x)$ .

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

One to any power is one.

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Separate fractions.

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

Simplify.

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Step 4.11.1

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

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

Step 4.11.2

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

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

Simplify.

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Step 4.14.1

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

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

Step 4.14.2

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

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

Step 4.15

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

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Step 4.15.1

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

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

Step 4.15.2

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

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

Step 4.15.3

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

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

Step 4.15.4

Add $1$ and $1$ .

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

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