Calculus Examples

$\tan (\cos (x)) + 12$

Step 1

By the Sum Rule, the derivative of $\tan (\cos (x)) + 12$ with respect to $x$ is $\frac{d}{d x} [\tan (\cos (x))] + \frac{d}{d x} [12]$ .

$\frac{d}{d x} [\tan (\cos (x))] + \frac{d}{d x} [12]$

Step 2

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

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Step 2.1

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 2.1.1

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

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [12]$

Step 2.1.2

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

$\sec^{2} (u) \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [12]$

Step 2.1.3

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

$\sec^{2} (\cos (x)) \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [12]$

Step 2.2

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

$\sec^{2} (\cos (x)) (- \sin (x)) + \frac{d}{d x} [12]$

Step 3

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$\sec^{2} (\cos (x)) (- \sin (x)) + 0$

Step 4

Simplify.

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Step 4.1

Add $\sec^{2} (\cos (x)) (- \sin (x))$ and $0$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

One to any power is one.

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Separate fractions.

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

Step 4.9

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

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

Step 4.10

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 4.11

Simplify.

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Step 4.11.1

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

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

Step 4.11.2

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

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

Step 4.12

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

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

Step 4.13

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

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Step 4.13.1

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

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

Step 4.13.2

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

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

Step 4.13.3

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

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

Step 4.13.4

Add $1$ and $1$ .

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

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