Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

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

Step 1.3

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Rewrite $\sec (\sin (x^{2}))$ in terms of sines and cosines.

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

Step 4.3

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

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

Step 4.4

One to any power is one.

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.6

Move $2$ to the left of $x$ .

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

Separate fractions.

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

Step 4.10

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

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

Step 4.11

Write $\cos (x^{2})$ as a fraction with denominator $1$ .

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

Step 4.12

Simplify.

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Step 4.12.1

Divide $\cos (x^{2})$ by $1$ .

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

Step 4.12.2

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

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

Step 4.13

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

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Step 4.13.1

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

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

Step 4.13.2

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

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

Step 4.14

Separate fractions.

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

Step 4.15

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

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

Step 4.16

Write $\cos (x^{2})$ as a fraction with denominator $1$ .

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

Step 4.17

Simplify.

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Step 4.17.1

Divide $\cos (x^{2})$ by $1$ .

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

Step 4.17.2

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

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

Step 4.18

Divide $(2 x) \sec (\sin (x^{2}))$ by $1$ .

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

Step 4.19

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

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Step 4.19.1

Raise $\sec (\sin (x^{2}))$ to the power of $1$ .

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

Step 4.19.2

Raise $\sec (\sin (x^{2}))$ to the power of $1$ .

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

Step 4.19.3

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

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

Step 4.19.4

Add $1$ and $1$ .

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