Calculus Examples

$y = \tan^{2} (\sin (x))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\tan^{2} (\sin (x)))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \tan (\sin (x))$ .

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Step 3.1.1

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

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

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

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

Step 3.1.3

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

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

Step 3.2

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 $g (x) = \sin (x)$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

One to any power is one.

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

Step 3.4.5

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

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

Step 3.4.6

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

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

Step 3.4.7

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

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

Step 3.4.8

Combine.

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

Step 3.4.9

Multiply $\cos^{2} (\sin (x))$ by $\cos (\sin (x))$ by adding the exponents.

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Step 3.4.9.1

Multiply $\cos^{2} (\sin (x))$ by $\cos (\sin (x))$ .

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Step 3.4.9.1.1

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

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

Step 3.4.9.1.2

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

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

Step 3.4.9.2

Add $2$ and $1$ .

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

Step 3.4.10

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

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

Step 3.4.11

Separate fractions.

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

Step 3.4.12

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

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

Step 3.4.13

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

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

Step 3.4.14

Simplify.

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Step 3.4.14.1

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

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

Step 3.4.14.2

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

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

Step 3.4.15

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

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

Step 3.4.16

Separate fractions.

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

Step 3.4.17

Convert from $\frac{\sin (\sin (x))}{\cos (\sin (x))}$ to $\tan (\sin (x))$ .

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

Step 3.4.18

Separate fractions.

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

Step 3.4.19

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

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

Step 3.4.20

Divide $2$ by $1$ .

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

Step 3.4.21

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

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Step 3.4.21.1

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

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

Step 3.4.21.2

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

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

Step 3.4.21.3

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

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

Step 3.4.21.4

Add $1$ and $1$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 2 \sec^{2} (\sin (x)) \tan (\sin (x)) \cos (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 2 \sec^{2} (\sin (x)) \tan (\sin (x)) \cos (x)$