Calculus Examples

$y = \sin (\tan (7 x))$

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 $g (x) = \tan (7 x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\tan (7 x)$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [\tan (7 x)]$

Step 1.2

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

$\cos (u_{1}) \frac{d}{d x} [\tan (7 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\tan (7 x)$ .

$\cos (\tan (7 x)) \frac{d}{d x} [\tan (7 x)]$

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 $g (x) = 7 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$\cos (\tan (7 x)) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [7 x])$

Step 2.2

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

$\cos (\tan (7 x)) (\sec^{2} (u_{2}) \frac{d}{d x} [7 x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$\cos (\tan (7 x)) (\sec^{2} (7 x) \frac{d}{d x} [7 x])$

Step 3

Differentiate.

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Step 3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\cos (\tan (7 x)) (\sec^{2} (7 x) (7 \frac{d}{d x} [x]))$

Step 3.2

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

$\cos (\tan (7 x)) (\sec^{2} (7 x) (7 \cdot 1))$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $7$ by $1$ .

$\cos (\tan (7 x)) (\sec^{2} (7 x) \cdot 7)$

Step 3.3.2

Move $7$ to the left of $\sec^{2} (7 x)$ .

$\cos (\tan (7 x)) (7 \cdot \sec^{2} (7 x))$

Step 4

Simplify.

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Step 4.1

Move $7$ to the left of $\cos (\tan (7 x))$ .

$7 \cos (\tan (7 x)) \sec^{2} (7 x)$

Step 4.2

Reorder the factors of $7 \cos (\tan (7 x)) \sec^{2} (7 x)$ .

$7 \sec^{2} (7 x) \cos (\tan (7 x))$

Step 4.3

Rewrite $\sec (7 x)$ in terms of sines and cosines.

$7 {(\frac{1}{\cos (7 x)})}^{2} \cos (\tan (7 x))$

Step 4.4

Apply the product rule to $\frac{1}{\cos (7 x)}$ .

$7 \frac{1^{2}}{\cos^{2} (7 x)} \cos (\tan (7 x))$

Step 4.5

One to any power is one.

$7 \frac{1}{\cos^{2} (7 x)} \cos (\tan (7 x))$

Step 4.6

Combine $7$ and $\frac{1}{\cos^{2} (7 x)}$ .

$\frac{7}{\cos^{2} (7 x)} \cos (\tan (7 x))$

Step 4.7

Combine $\frac{7}{\cos^{2} (7 x)}$ and $\cos (\tan (7 x))$ .

$\frac{7 \cos (\tan (7 x))}{\cos^{2} (7 x)}$

Step 4.8

Factor $\cos (7 x)$ out of $\cos^{2} (7 x)$ .

$\frac{7 \cos (\tan (7 x))}{\cos (7 x) \cos (7 x)}$

Step 4.9

Separate fractions.

$\frac{7}{\cos (7 x)} \cdot \frac{\cos (\tan (7 x))}{\cos (7 x)}$

Step 4.10

Rewrite $\frac{\cos (\tan (7 x))}{\cos (7 x)}$ as a product.

$\frac{7}{\cos (7 x)} (\cos (\tan (7 x)) \frac{1}{\cos (7 x)})$

Step 4.11

Write $\cos (\tan (7 x))$ as a fraction with denominator $1$ .

$\frac{7}{\cos (7 x)} (\frac{\cos (\tan (7 x))}{1} \cdot \frac{1}{\cos (7 x)})$

Step 4.12

Simplify.

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Step 4.12.1

Divide $\cos (\tan (7 x))$ by $1$ .

$\frac{7}{\cos (7 x)} (\cos (\tan (7 x)) \frac{1}{\cos (7 x)})$

Step 4.12.2

Convert from $\frac{1}{\cos (7 x)}$ to $\sec (7 x)$ .

$\frac{7}{\cos (7 x)} (\cos (\tan (7 x)) \sec (7 x))$

Step 4.13

Separate fractions.

$\frac{7}{1} \cdot \frac{1}{\cos (7 x)} (\cos (\tan (7 x)) \sec (7 x))$

Step 4.14

Convert from $\frac{1}{\cos (7 x)}$ to $\sec (7 x)$ .

$\frac{7}{1} \sec (7 x) (\cos (\tan (7 x)) \sec (7 x))$

Step 4.15

Divide $7$ by $1$ .

$7 \sec (7 x) (\cos (\tan (7 x)) \sec (7 x))$

Step 4.16

Multiply $7 \sec (7 x) (\cos (\tan (7 x)) \sec (7 x))$ .

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Step 4.16.1

Raise $\sec (7 x)$ to the power of $1$ .

$7 (\sec^{1} (7 x) \sec (7 x)) \cos (\tan (7 x))$

Step 4.16.2

Raise $\sec (7 x)$ to the power of $1$ .

$7 (\sec^{1} (7 x) \sec^{1} (7 x)) \cos (\tan (7 x))$

Step 4.16.3

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

$7 {\sec (7 x)}^{1 + 1} \cos (\tan (7 x))$

Step 4.16.4

Add $1$ and $1$ .

$7 \sec^{2} (7 x) \cos (\tan (7 x))$