Calculus Examples

$\cot^{2} (\cos (x))$

Step 1

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

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cot (\cos (x))$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\cot (\cos (x))$ .

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

The derivative of $\cot (u_{2})$ with respect to $u_{2}$ is $- \csc^{2} (u_{2})$ .

$2 \cot (\cos (x)) (- \csc^{2} (u_{2}) \frac{d}{d x} [\cos (x)])$

Step 2.3

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

$2 \cot (\cos (x)) (- \csc^{2} (\cos (x)) \frac{d}{d x} [\cos (x)])$

Step 3

Multiply $- 1$ by $2$ .

$- 2 \cot (\cos (x)) (\csc^{2} (\cos (x)) \frac{d}{d x} [\cos (x)])$

Step 4

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

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

Step 5

Multiply $- 1$ by $- 2$ .

$2 \cot (\cos (x)) \csc^{2} (\cos (x)) \sin (x)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$2 \csc^{2} (\cos (x)) \cot (\cos (x)) \sin (x)$

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

One to any power is one.

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

Combine.

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

Step 6.8

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

Tap for more steps...

Step 6.8.1

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

Tap for more steps...

Step 6.8.1.1

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

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

Step 6.8.1.2

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

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

Step 6.8.2

Add $2$ and $1$ .

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

Step 6.9

Combine $\frac{2 \cos (\cos (x))}{\sin^{3} (\cos (x))}$ and $\sin (x)$ .

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

Step 6.10

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

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

Step 6.11

Separate fractions.

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

Step 6.12

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

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

Step 6.13

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

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

Step 6.14

Separate fractions.

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

Step 6.15

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

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

Step 6.16

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

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

Step 6.17

Simplify.

Tap for more steps...

Step 6.17.1

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

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

Step 6.17.2

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

$\frac{2}{\sin (\cos (x))} (\sin (x) \csc (\cos (x))) \cot (\cos (x))$

Step 6.18

Separate fractions.

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

Step 6.19

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

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

Step 6.20

Divide $2$ by $1$ .

$2 \csc (\cos (x)) (\sin (x) \csc (\cos (x))) \cot (\cos (x))$

Step 6.21

Multiply $2 \csc (\cos (x)) (\sin (x) \csc (\cos (x)))$ .

Tap for more steps...

Step 6.21.1

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

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

Step 6.21.2

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

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

Step 6.21.3

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

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

Step 6.21.4

Add $1$ and $1$ .

$2 \csc^{2} (\cos (x)) \sin (x) \cot (\cos (x))$