Calculus Examples

$y = \sin (\sec (x))$

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 1.2

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

$\cos (u) \frac{d}{d x} [\sec (x)]$

Step 1.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.5

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

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

Step 3.6

Separate fractions.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Simplify.

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Step 3.9.1

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

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

Step 3.9.2

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

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

Step 3.10

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

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