Calculus Examples

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Add $1$ and $1$ .

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

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Reorder terms.

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

Step 14.3

Simplify each term.

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Step 14.3.1

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

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

Step 14.3.2

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

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

Step 14.3.3

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

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Step 14.3.3.1

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

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

Step 14.3.3.2

Cancel the common factor.

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

Step 14.3.3.3

Rewrite the expression.

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

Step 14.3.4

One to any power is one.

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

Step 14.3.5

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

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

Step 14.3.6

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

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

Step 14.3.7

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

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Step 14.3.7.1

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

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

Step 14.3.7.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 14.3.7.2.1

Multiply $\sin (x)$ by $\sin^{2} (x)$ .

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Step 14.3.7.2.1.1

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

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

Step 14.3.7.2.1.2

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

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

Step 14.3.7.2.2

Add $1$ and $2$ .

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

Step 14.3.8

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

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

Step 14.3.9

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

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Step 14.3.9.1

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

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

Step 14.3.9.2

Cancel the common factor.

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

Step 14.3.9.3

Rewrite the expression.

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

Step 14.4

Simplify each term.

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Step 14.4.1

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

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

Step 14.4.2

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

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

Step 14.4.3

Separate fractions.

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

Step 14.4.4

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

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

Step 14.4.5

Divide $\sin^{2} (x)$ by $1$ .

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