Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $2 \frac{\sin (x)}{\cos (x)} + 2 \cot (x) \cdot \sin (x)$ with respect to $x$ is $\frac{d}{d x} [2 \frac{\sin (x)}{\cos (x)}] + \frac{d}{d x} [2 \cot (x) \cdot \sin (x)]$ .

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

Step 2

Evaluate $\frac{d}{d x} [2 \frac{\sin (x)}{\cos (x)}]$ .

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Step 2.1

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

$\frac{d}{d x} [2 \tan (x)] + \frac{d}{d x} [2 \cot (x) \cdot \sin (x)]$

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 \tan (x)$ with respect to $x$ is $2 \frac{d}{d x} [\tan (x)]$ .

$2 \frac{d}{d x} [\tan (x)] + \frac{d}{d x} [2 \cot (x) \cdot \sin (x)]$

Step 2.3

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$2 \sec^{2} (x) + \frac{d}{d x} [2 \cot (x) \cdot \sin (x)]$

Step 3

Evaluate $\frac{d}{d x} [2 \cot (x) \cdot \sin (x)]$ .

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \cot (x) \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [\cot (x) \sin (x)]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $2$ .

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

Step 4.3

Reorder terms.

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

Step 4.4

Simplify each term.

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Step 4.4.1

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

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

Step 4.4.2

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

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Step 4.4.2.1

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

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

Step 4.4.2.2

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

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

Step 4.4.2.3

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

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

Step 4.4.2.4

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

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

Step 4.4.2.5

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

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

Step 4.4.2.6

Add $1$ and $1$ .

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

Step 4.4.3

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

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

Step 4.4.4

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

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

Step 4.4.5

One to any power is one.

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

Step 4.4.6

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

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

Step 4.4.7

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

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Step 4.4.7.1

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

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

Step 4.4.7.2

Cancel the common factor.

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

Step 4.4.7.3

Rewrite the expression.

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

Step 4.4.8

Move the negative in front of the fraction.

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

Step 4.4.9

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

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

Step 4.4.10

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

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

Step 4.4.11

One to any power is one.

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

Step 4.4.12

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

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

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

Step 4.7

Factor $- 2$ out of $- 2$ .

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

Apply pythagorean identity.

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

Step 4.11

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

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Step 4.11.1

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

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

Step 4.11.2

Cancel the common factors.

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Step 4.11.2.1

Multiply by $1$ .

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

Step 4.11.2.2

Cancel the common factor.

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

Step 4.11.2.3

Rewrite the expression.

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

Step 4.11.2.4

Divide $- 2 \sin (x)$ by $1$ .

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

Step 4.12

Simplify each term.

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Step 4.12.1

Multiply by $1$ .

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

Step 4.12.2

Separate fractions.

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

Step 4.12.3

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

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

Step 4.12.4

Divide $2$ by $1$ .

$- 2 \sin (x) + 2 \sec^{2} (x)$