Calculus Examples

$y = 4 \sin (x) \tan (x)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \sin (x) \tan (x)$ with respect to $x$ is $4 \frac{d}{d x} [\sin (x) \tan (x)]$ .

$4 \frac{d}{d x} [\sin (x) \tan (x)]$

Step 2

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

$4 (\sin (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sin (x)])$

Step 3

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

$4 (\sin (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sin (x)])$

Step 4

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

$4 (\sin (x) \sec^{2} (x) + \tan (x) \cos (x))$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

$4 (\sin (x) \sec^{2} (x)) + 4 (\tan (x) \cos (x))$

Step 5.2

Remove parentheses.

$4 \sin (x) \sec^{2} (x) + 4 \tan (x) \cos (x)$

Step 5.3

Reorder terms.

$4 \sec^{2} (x) \sin (x) + 4 \cos (x) \tan (x)$

Step 5.4

Simplify each term.

Tap for more steps...

Step 5.4.1

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

$4 {(\frac{1}{\cos (x)})}^{2} \sin (x) + 4 \cos (x) \tan (x)$

Step 5.4.2

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

$4 \frac{1^{2}}{\cos^{2} (x)} \sin (x) + 4 \cos (x) \tan (x)$

Step 5.4.3

One to any power is one.

$4 \frac{1}{\cos^{2} (x)} \sin (x) + 4 \cos (x) \tan (x)$

Step 5.4.4

Combine $4$ and $\frac{1}{\cos^{2} (x)}$ .

$\frac{4}{\cos^{2} (x)} \sin (x) + 4 \cos (x) \tan (x)$

Step 5.4.5

Combine $\frac{4}{\cos^{2} (x)}$ and $\sin (x)$ .

$\frac{4 \sin (x)}{\cos^{2} (x)} + 4 \cos (x) \tan (x)$

Step 5.4.6

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

Tap for more steps...

Step 5.4.6.1

Add parentheses.

$\frac{4 \sin (x)}{\cos^{2} (x)} + 4 (\cos (x) \tan (x))$

Step 5.4.6.2

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

$\frac{4 \sin (x)}{\cos^{2} (x)} + 4 (\tan (x) \cos (x))$

Step 5.4.6.3

Rewrite $4 \cos (x) \tan (x)$ in terms of sines and cosines.

$\frac{4 \sin (x)}{\cos^{2} (x)} + 4 (\frac{\sin (x)}{\cos (x)} \cos (x))$

Step 5.4.6.4

Cancel the common factors.

$\frac{4 \sin (x)}{\cos^{2} (x)} + 4 \sin (x)$

Step 5.5

Simplify each term.

Tap for more steps...

Step 5.5.1

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

$\frac{4 \sin (x)}{\cos (x) \cos (x)} + 4 \sin (x)$

Step 5.5.2

Separate fractions.

$\frac{4}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + 4 \sin (x)$

Step 5.5.3

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

$\frac{4}{\cos (x)} \tan (x) + 4 \sin (x)$

Step 5.5.4

Separate fractions.

$\frac{4}{1} \cdot \frac{1}{\cos (x)} \tan (x) + 4 \sin (x)$

Step 5.5.5

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

$\frac{4}{1} \sec (x) \tan (x) + 4 \sin (x)$

Step 5.5.6

Divide $4$ by $1$ .

$4 \sec (x) \tan (x) + 4 \sin (x)$