Calculus Examples

$5 \cos (x) \cot (x)$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (x) \cot (x)$ with respect to $x$ is $5 \frac{d}{d x} [\cos (x) \cot (x)]$ .

$5 \frac{d}{d x} [\cos (x) \cot (x)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos (x)$ and $g (x) = \cot (x)$ .

$5 (\cos (x) \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [\cos (x)])$

Step 3

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

$5 (\cos (x) (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [\cos (x)])$

Step 4

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

$5 (\cos (x) (- \csc^{2} (x)) + \cot (x) (- \sin (x)))$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

$5 (\cos (x) (- \csc^{2} (x))) + 5 (\cot (x) (- \sin (x)))$

Step 5.2

Combine terms.

Tap for more steps...

Step 5.2.1

Multiply $- 1$ by $5$ .

$- 5 (\cos (x) (\csc^{2} (x))) + 5 (\cot (x) (- \sin (x)))$

Step 5.2.2

Multiply $- 1$ by $5$ .

$- 5 \cos (x) \csc^{2} (x) - 5 \cot (x) \sin (x)$

Step 5.3

Reorder terms.

$- 5 \csc^{2} (x) \cos (x) - 5 \cot (x) \sin (x)$

Step 5.4

Simplify each term.

Tap for more steps...

Step 5.4.1

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

$- 5 {(\frac{1}{\sin (x)})}^{2} \cos (x) - 5 \cot (x) \sin (x)$

Step 5.4.2

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

$- 5 \frac{1^{2}}{\sin^{2} (x)} \cos (x) - 5 \cot (x) \sin (x)$

Step 5.4.3

One to any power is one.

$- 5 \frac{1}{\sin^{2} (x)} \cos (x) - 5 \cot (x) \sin (x)$

Step 5.4.4

Combine $- 5$ and $\frac{1}{\sin^{2} (x)}$ .

$\frac{- 5}{\sin^{2} (x)} \cos (x) - 5 \cot (x) \sin (x)$

Step 5.4.5

Move the negative in front of the fraction.

$- \frac{5}{\sin^{2} (x)} \cos (x) - 5 \cot (x) \sin (x)$

Step 5.4.6

Combine $\cos (x)$ and $\frac{5}{\sin^{2} (x)}$ .

$- \frac{\cos (x) \cdot 5}{\sin^{2} (x)} - 5 \cot (x) \sin (x)$

Step 5.4.7

Move $5$ to the left of $\cos (x)$ .

$- \frac{5 \cdot \cos (x)}{\sin^{2} (x)} - 5 \cot (x) \sin (x)$

Step 5.4.8

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 5.4.8.1

Add parentheses.

$- \frac{5 \cos (x)}{\sin^{2} (x)} - 5 (\cot (x) \sin (x))$

Step 5.4.8.2

Rewrite $- 5 \cot (x) \sin (x)$ in terms of sines and cosines.

$- \frac{5 \cos (x)}{\sin^{2} (x)} - 5 (\frac{\cos (x)}{\sin (x)} \sin (x))$

Step 5.4.8.3

Cancel the common factors.

$- \frac{5 \cos (x)}{\sin^{2} (x)} - 5 \cos (x)$

Step 5.5

Simplify each term.

Tap for more steps...

Step 5.5.1

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

$- \frac{5 \cos (x)}{\sin (x) \sin (x)} - 5 \cos (x)$

Step 5.5.2

Separate fractions.

$- (\frac{5}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}) - 5 \cos (x)$

Step 5.5.3

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

$- (\frac{5}{\sin (x)} \cot (x)) - 5 \cos (x)$

Step 5.5.4

Separate fractions.

$- (\frac{5}{1} \cdot \frac{1}{\sin (x)} \cot (x)) - 5 \cos (x)$

Step 5.5.5

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

$- (\frac{5}{1} \csc (x) \cot (x)) - 5 \cos (x)$

Step 5.5.6

Divide $5$ by $1$ .

$- (5 \csc (x) \cot (x)) - 5 \cos (x)$

Step 5.5.7

Multiply $5$ by $- 1$ .

$- 5 \csc (x) \cot (x) - 5 \cos (x)$