Calculus Examples

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Cancel the common factors.

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

One to any power is one.

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

Step 4.2.5

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

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

Step 4.3

Simplify each term.

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Step 4.3.1

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

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

Step 4.3.2

Separate fractions.

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

Step 4.3.3

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

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

Step 4.3.4

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

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