Calculus Examples

$y = 3 \sin (x) \cot (x)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 \sin (x) \cot (x)$ with respect to $x$ is $3 \frac{d}{d x} [\sin (x) \cot (x)]$ .

$3 \frac{d}{d x} [\sin (x) \cot (x)]$

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

$3 (\sin (x) \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [\sin (x)])$

Step 3

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

$3 (\sin (x) (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [\sin (x)])$

Step 4

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

$3 (\sin (x) (- \csc^{2} (x)) + \cot (x) \cos (x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$3 (\sin (x) (- \csc^{2} (x))) + 3 (\cot (x) \cos (x))$

Step 5.2

Multiply $- 1$ by $3$ .

$- 3 \sin (x) \csc^{2} (x) + 3 \cot (x) \cos (x)$

Step 5.3

Reorder terms.

$- 3 \csc^{2} (x) \sin (x) + 3 \cos (x) \cot (x)$

Step 5.4

Simplify each term.

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Step 5.4.1

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

$- 3 {(\frac{1}{\sin (x)})}^{2} \sin (x) + 3 \cos (x) \cot (x)$

Step 5.4.2

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

$- 3 \frac{1^{2}}{\sin^{2} (x)} \sin (x) + 3 \cos (x) \cot (x)$

Step 5.4.3

One to any power is one.

$- 3 \frac{1}{\sin^{2} (x)} \sin (x) + 3 \cos (x) \cot (x)$

Step 5.4.4

Combine $- 3$ and $\frac{1}{\sin^{2} (x)}$ .

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

Step 5.4.5

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

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Step 5.4.5.1

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

$\frac{- 3}{\sin (x) \sin (x)} \sin (x) + 3 \cos (x) \cot (x)$

Step 5.4.5.2

Cancel the common factor.

$\frac{- 3}{\sin (x) \sin (x)} \sin (x) + 3 \cos (x) \cot (x)$

Step 5.4.5.3

Rewrite the expression.

$\frac{- 3}{\sin (x)} + 3 \cos (x) \cot (x)$

Step 5.4.6

Move the negative in front of the fraction.

$- \frac{3}{\sin (x)} + 3 \cos (x) \cot (x)$

Step 5.4.7

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

$- \frac{3}{\sin (x)} + 3 \cos (x) \frac{\cos (x)}{\sin (x)}$

Step 5.4.8

Multiply $3 \cos (x) \frac{\cos (x)}{\sin (x)}$ .

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Step 5.4.8.1

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

$- \frac{3}{\sin (x)} + \frac{\cos (x) \cdot 3}{\sin (x)} \cos (x)$

Step 5.4.8.2

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

$- \frac{3}{\sin (x)} + \frac{\cos (x) \cdot 3 \cos (x)}{\sin (x)}$

Step 5.4.8.3

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

$- \frac{3}{\sin (x)} + \frac{3 (\cos^{1} (x) \cos (x))}{\sin (x)}$

Step 5.4.8.4

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

$- \frac{3}{\sin (x)} + \frac{3 (\cos^{1} (x) \cos^{1} (x))}{\sin (x)}$

Step 5.4.8.5

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

$- \frac{3}{\sin (x)} + \frac{3 {\cos (x)}^{1 + 1}}{\sin (x)}$

Step 5.4.8.6

Add $1$ and $1$ .

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Factor $- 3$ out of $- 3$ .

$\frac{- 3 (1) + 3 \cos^{2} (x)}{\sin (x)}$

Step 5.7

Factor $- 3$ out of $3 \cos^{2} (x)$ .

$\frac{- 3 (1) - 3 (- \cos^{2} (x))}{\sin (x)}$

Step 5.8

Factor $- 3$ out of $- 3 (1) - 3 (- \cos^{2} (x))$ .

$\frac{- 3 (1 - \cos^{2} (x))}{\sin (x)}$

Step 5.9

Apply pythagorean identity.

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

Step 5.10

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

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Step 5.10.1

Factor $\sin (x)$ out of $- 3 \sin^{2} (x)$ .

$\frac{\sin (x) (- 3 \sin (x))}{\sin (x)}$

Step 5.10.2

Cancel the common factors.

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Step 5.10.2.1

Multiply by $1$ .

$\frac{\sin (x) (- 3 \sin (x))}{\sin (x) \cdot 1}$

Step 5.10.2.2

Cancel the common factor.

$\frac{\sin (x) (- 3 \sin (x))}{\sin (x) \cdot 1}$

Step 5.10.2.3

Rewrite the expression.

$\frac{- 3 \sin (x)}{1}$

Step 5.10.2.4

Divide $- 3 \sin (x)$ by $1$ .

$- 3 \sin (x)$