Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

One to any power is one.

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

Step 4.2.4

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

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

Step 4.2.5

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

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Step 4.2.5.1

Add parentheses.

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

Step 4.2.5.2

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

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

Step 4.2.5.3

Cancel the common factors.

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

Step 4.3

Simplify each term.

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Step 4.3.1

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

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

Step 4.3.2

Separate fractions.

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

Step 4.3.3

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

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

Step 4.3.4

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

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