Calculus Examples

$y = 2 \cos (x) \tan (x)$

Step 1

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

$2 \frac{d}{d x} [\cos (x) \tan (x)]$

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

$2 (\cos (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\cos (x)])$

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $2$ .

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

Step 5.3

Reorder terms.

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

Step 5.4

Simplify each term.

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

One to any power is one.

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

Step 5.4.4

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

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

Step 5.4.5

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

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Step 5.4.5.1

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

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

Step 5.4.5.2

Cancel the common factor.

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

Step 5.4.5.3

Rewrite the expression.

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

Step 5.4.6

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

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

Step 5.4.7

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

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Step 5.4.7.1

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

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

Step 5.4.7.2

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

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

Step 5.4.7.3

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

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

Step 5.4.7.4

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

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

Step 5.4.7.5

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

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

Step 5.4.7.6

Add $1$ and $1$ .

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

Step 5.4.8

Move the negative in front of the fraction.

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Factor $2$ out of $2$ .

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

Apply pythagorean identity.

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

Step 5.10

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

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Step 5.10.1

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

$\frac{\cos (x) (2 \cos (x))}{\cos (x)}$

Step 5.10.2

Cancel the common factors.

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Step 5.10.2.1

Multiply by $1$ .

$\frac{\cos (x) (2 \cos (x))}{\cos (x) \cdot 1}$

Step 5.10.2.2

Cancel the common factor.

$\frac{\cos (x) (2 \cos (x))}{\cos (x) \cdot 1}$

Step 5.10.2.3

Rewrite the expression.

$\frac{2 \cos (x)}{1}$

Step 5.10.2.4

Divide $2 \cos (x)$ by $1$ .

$2 \cos (x)$