Calculus Examples

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

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

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

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

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

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 $f (x) = x^{2}$ and $g (x) = \tan (x)$ .

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

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

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

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

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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

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

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

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

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

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

One to any power is one.

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

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

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

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

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

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

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

Combine.

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

Separate fractions.

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

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

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

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

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

Simplify.

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

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

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

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

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

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

Separate fractions.

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

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

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

Separate fractions.

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

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

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

Divide $2$ by $1$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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