Calculus Examples

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

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

Step 4.2.2.4

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

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

Step 4.2.2.5

Add $1$ and $1$ .

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

Cancel the common factor.

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

Step 4.2.5.3

Rewrite the expression.

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

Step 4.2.6

One to any power is one.

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

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

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

Step 4.5

Apply pythagorean identity.

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Multiply by $1$ .

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

Step 4.6.2.2

Cancel the common factor.

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

Step 4.6.2.3

Rewrite the expression.

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

Step 4.6.2.4

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

$\cos (x)$