Calculus Examples

$f (t) = \sin (t) \tan (t)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = \sin (t)$ and $g (t) = \tan (t)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

One to any power is one.

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

Step 4.2.4

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

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

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.5.3

Cancel the common factors.

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

Step 4.3

Simplify each term.

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Step 4.3.1

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

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

Step 4.3.2

Separate fractions.

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

Step 4.3.3

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

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

Step 4.3.4

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

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