Calculus Examples

$f (x) = \cos (\tan (3 x))$

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 $g (x) = \tan (3 x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\tan (3 x)$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [\tan (3 x)]$

Step 1.2

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

$- \sin (u_{1}) \frac{d}{d x} [\tan (3 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\tan (3 x)$ .

$- \sin (\tan (3 x)) \frac{d}{d x} [\tan (3 x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$- \sin (\tan (3 x)) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [3 x])$

Step 2.2

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

$- \sin (\tan (3 x)) (\sec^{2} (u_{2}) \frac{d}{d x} [3 x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$- \sin (\tan (3 x)) (\sec^{2} (3 x) \frac{d}{d x} [3 x])$

Step 3

Differentiate.

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- \sin (\tan (3 x)) \sec^{2} (3 x) (3 \frac{d}{d x} [x])$

Step 3.2

Multiply $3$ by $- 1$ .

$- 3 \sin (\tan (3 x)) \sec^{2} (3 x) \frac{d}{d x} [x]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 3 \sin (\tan (3 x)) \sec^{2} (3 x) \cdot 1$

Step 3.4

Multiply $- 3$ by $1$ .

$- 3 \sin (\tan (3 x)) \sec^{2} (3 x)$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $- 3 \sin (\tan (3 x)) \sec^{2} (3 x)$ .

$- 3 \sec^{2} (3 x) \sin (\tan (3 x))$

Step 4.2

Rewrite $\sec (3 x)$ in terms of sines and cosines.

$- 3 {(\frac{1}{\cos (3 x)})}^{2} \sin (\tan (3 x))$

Step 4.3

Apply the product rule to $\frac{1}{\cos (3 x)}$ .

$- 3 \frac{1^{2}}{\cos^{2} (3 x)} \sin (\tan (3 x))$

Step 4.4

One to any power is one.

$- 3 \frac{1}{\cos^{2} (3 x)} \sin (\tan (3 x))$

Step 4.5

Combine $- 3$ and $\frac{1}{\cos^{2} (3 x)}$ .

$\frac{- 3}{\cos^{2} (3 x)} \sin (\tan (3 x))$

Step 4.6

Move the negative in front of the fraction.

$- \frac{3}{\cos^{2} (3 x)} \sin (\tan (3 x))$

Step 4.7

Combine $\sin (\tan (3 x))$ and $\frac{3}{\cos^{2} (3 x)}$ .

$- \frac{\sin (\tan (3 x)) \cdot 3}{\cos^{2} (3 x)}$

Step 4.8

Move $3$ to the left of $\sin (\tan (3 x))$ .

$- \frac{3 \cdot \sin (\tan (3 x))}{\cos^{2} (3 x)}$

Step 4.9

Factor $\cos (3 x)$ out of $\cos^{2} (3 x)$ .

$- \frac{3 \sin (\tan (3 x))}{\cos (3 x) \cos (3 x)}$

Step 4.10

Separate fractions.

$- (\frac{3}{\cos (3 x)} \cdot \frac{\sin (\tan (3 x))}{\cos (3 x)})$

Step 4.11

Rewrite $\frac{\sin (\tan (3 x))}{\cos (3 x)}$ as a product.

$- (\frac{3}{\cos (3 x)} (\sin (\tan (3 x)) \frac{1}{\cos (3 x)}))$

Step 4.12

Write $\sin (\tan (3 x))$ as a fraction with denominator $1$ .

$- (\frac{3}{\cos (3 x)} (\frac{\sin (\tan (3 x))}{1} \cdot \frac{1}{\cos (3 x)}))$

Step 4.13

Simplify.

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Step 4.13.1

Divide $\sin (\tan (3 x))$ by $1$ .

$- (\frac{3}{\cos (3 x)} (\sin (\tan (3 x)) \frac{1}{\cos (3 x)}))$

Step 4.13.2

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

$- (\frac{3}{\cos (3 x)} (\sin (\tan (3 x)) \sec (3 x)))$

Step 4.14

Separate fractions.

$- (\frac{3}{1} \cdot \frac{1}{\cos (3 x)} (\sin (\tan (3 x)) \sec (3 x)))$

Step 4.15

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

$- (\frac{3}{1} \sec (3 x) (\sin (\tan (3 x)) \sec (3 x)))$

Step 4.16

Divide $3$ by $1$ .

$- (3 \sec (3 x) (\sin (\tan (3 x)) \sec (3 x)))$

Step 4.17

Multiply $3 \sec (3 x) (\sin (\tan (3 x)) \sec (3 x))$ .

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Step 4.17.1

Raise $\sec (3 x)$ to the power of $1$ .

$- (3 (\sec^{1} (3 x) \sec (3 x)) \sin (\tan (3 x)))$

Step 4.17.2

Raise $\sec (3 x)$ to the power of $1$ .

$- (3 (\sec^{1} (3 x) \sec^{1} (3 x)) \sin (\tan (3 x)))$

Step 4.17.3

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

$- (3 {\sec (3 x)}^{1 + 1} \sin (\tan (3 x)))$

Step 4.17.4

Add $1$ and $1$ .

$- (3 \sec^{2} (3 x) \sin (\tan (3 x)))$

Step 4.18

Multiply $3$ by $- 1$ .

$- 3 \sec^{2} (3 x) \sin (\tan (3 x))$