Calculus Examples

$f (x) = \tan (x) \cdot (\sin (x) + \cos (x))$

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 $f (x) = \tan (x)$ and $g (x) = \sin (x) + \cos (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Reorder terms.

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

Step 6.4

Simplify each term.

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Step 6.4.1

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

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Step 6.4.1.1

Reorder $\cos (x)$ and $\tan (x)$ .

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

Step 6.4.1.2

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

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

Step 6.4.1.3

Cancel the common factors.

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

Step 6.4.2

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

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

Step 6.4.3

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

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Step 6.4.3.1

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

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

Step 6.4.3.2

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

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

Step 6.4.3.3

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

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

Step 6.4.3.4

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

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

Step 6.4.3.5

Add $1$ and $1$ .

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

Step 6.4.4

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

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

Step 6.4.5

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

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

Step 6.4.6

One to any power is one.

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

Step 6.4.7

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

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

Step 6.4.8

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

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

Step 6.4.9

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

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

Step 6.4.10

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

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Step 6.4.10.1

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

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

Step 6.4.10.2

Cancel the common factor.

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

Step 6.4.10.3

Rewrite the expression.

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

Step 6.4.11

One to any power is one.

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

Step 6.5

Combine the numerators over the common denominator.

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

Step 6.6

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

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

Step 6.7

Apply pythagorean identity.

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

Step 6.8

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

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Step 6.8.1

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

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

Step 6.8.2

Cancel the common factors.

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Step 6.8.2.1

Multiply by $1$ .

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

Step 6.8.2.2

Cancel the common factor.

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

Step 6.8.2.3

Rewrite the expression.

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

Step 6.8.2.4

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

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

Step 6.9

Simplify each term.

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Step 6.9.1

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

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

Step 6.9.2

Separate fractions.

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

Step 6.9.3

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

$\sin (x) + \cos (x) + \frac{1}{\cos (x)} \tan (x)$

Step 6.9.4

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

$\sin (x) + \cos (x) + \sec (x) \tan (x)$