Calculus Examples

$f (x) = \sin (\tan (2 x))$

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 $g (x) = \tan (2 x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\tan (2 x)$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [\tan (2 x)]$

Step 1.2

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

$\cos (u_{1}) \frac{d}{d x} [\tan (2 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\tan (2 x)$ .

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

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 $g (x) = 2 x$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 3.2

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

$\cos (\tan (2 x)) \sec^{2} (2 x) (2 \cdot 1)$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $2$ by $1$ .

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

Step 3.3.2

Move $2$ to the left of $\cos (\tan (2 x)) \sec^{2} (2 x)$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Rewrite $\sec (2 x)$ in terms of sines and cosines.

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

Step 4.3

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

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

Step 4.4

One to any power is one.

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

$\frac{2 \cos (\tan (2 x))}{\cos (2 x) \cos (2 x)}$

Step 4.8

Separate fractions.

$\frac{2}{\cos (2 x)} \cdot \frac{\cos (\tan (2 x))}{\cos (2 x)}$

Step 4.9

Rewrite $\frac{\cos (\tan (2 x))}{\cos (2 x)}$ as a product.

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

Step 4.10

Write $\cos (\tan (2 x))$ as a fraction with denominator $1$ .

$\frac{2}{\cos (2 x)} (\frac{\cos (\tan (2 x))}{1} \cdot \frac{1}{\cos (2 x)})$

Step 4.11

Simplify.

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Step 4.11.1

Divide $\cos (\tan (2 x))$ by $1$ .

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

Step 4.11.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\frac{2}{\cos (2 x)} (\cos (\tan (2 x)) \sec (2 x))$

Step 4.12

Separate fractions.

$\frac{2}{1} \cdot \frac{1}{\cos (2 x)} (\cos (\tan (2 x)) \sec (2 x))$

Step 4.13

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\frac{2}{1} \sec (2 x) (\cos (\tan (2 x)) \sec (2 x))$

Step 4.14

Divide $2$ by $1$ .

$2 \sec (2 x) (\cos (\tan (2 x)) \sec (2 x))$

Step 4.15

Multiply $2 \sec (2 x) (\cos (\tan (2 x)) \sec (2 x))$ .

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Step 4.15.1

Raise $\sec (2 x)$ to the power of $1$ .

$2 (\sec^{1} (2 x) \sec (2 x)) \cos (\tan (2 x))$

Step 4.15.2

Raise $\sec (2 x)$ to the power of $1$ .

$2 (\sec^{1} (2 x) \sec^{1} (2 x)) \cos (\tan (2 x))$

Step 4.15.3

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

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

Step 4.15.4

Add $1$ and $1$ .

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