Calculus Examples

$\cot (x) \sin (x)$

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 $f (x) = \cot (x)$ and $g (x) = \sin (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

Multiply $\cos (x) \frac{\cos (x)}{\sin (x)}$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

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

Step 4.2.2.4

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

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

Step 4.2.2.5

Add $1$ and $1$ .

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Cancel the common factor of $\sin (x)$ .

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Step 4.2.5.1

Move the leading negative in $- \frac{1^{2}}{\sin^{2} (x)}$ into the numerator.

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

Step 4.2.5.2

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

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

Step 4.2.5.3

Cancel the common factor.

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

Step 4.2.5.4

Rewrite the expression.

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

Step 4.2.6

One to any power is one.

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

Step 4.2.7

Multiply $- 1$ by $1$ .

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

Step 4.2.8

Move the negative in front of the fraction.

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 4.5

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.6

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 4.7

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 4.8

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 4.9

Apply pythagorean identity.

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

Step 4.10

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 4.10.1

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

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

Step 4.10.2

Cancel the common factors.

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Step 4.10.2.1

Multiply by $1$ .

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

Step 4.10.2.2

Cancel the common factor.

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

Step 4.10.2.3

Rewrite the expression.

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

Step 4.10.2.4

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

$- \sin (x)$