Calculus Examples

$y = \tan (\cos (x))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\tan (\cos (x)))$

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 $f (x) = \tan (x)$ and $g (x) = \cos (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

Simplify.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

One to any power is one.

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

Separate fractions.

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

Simplify.

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Step 3.3.10.1

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

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

Step 3.3.10.2

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

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

Step 3.3.11

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

$- (\sec (\cos (x)) (\sin (x) \sec (\cos (x))))$

Step 3.3.12

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

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Step 3.3.12.1

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

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

Step 3.3.12.2

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

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

Step 3.3.12.3

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

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

Step 3.3.12.4

Add $1$ and $1$ .

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

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

Step 4

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

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

Step 5

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

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