Calculus Examples

$y = 6 x^{2} \sin (x) \tan (x)$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2} \sin (x) \tan (x)$ with respect to $x$ is $6 \frac{d}{d x} [x^{2} \sin (x) \tan (x)]$ .

$6 \frac{d}{d x} [x^{2} \sin (x) \tan (x)]$

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 $f (x) = x^{2} \sin (x)$ and $g (x) = \tan (x)$ .

$6 (x^{2} \sin (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [x^{2} \sin (x)])$

Step 3

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

$6 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [x^{2} \sin (x)])$

Step 4

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 $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

$6 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{2}]))$

Step 5

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

$6 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \cos (x) + \sin (x) \frac{d}{d x} [x^{2}]))$

Step 6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$6 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \cos (x) + \sin (x) (2 x)))$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Apply the distributive property.

$6 (x^{2} \sin (x) \sec^{2} (x) + \tan (x) (x^{2} \cos (x)) + \tan (x) (\sin (x) (2 x)))$

Step 7.2

Apply the distributive property.

$6 (x^{2} \sin (x) \sec^{2} (x)) + 6 (\tan (x) (x^{2} \cos (x))) + 6 (\tan (x) (\sin (x) (2 x)))$

Step 7.3

Multiply $2$ by $6$ .

$6 x^{2} \sin (x) \sec^{2} (x) + 6 \tan (x) x^{2} \cos (x) + 12 \tan (x) \sin (x) x$

Step 7.4

Reorder terms.

$6 x^{2} \sec^{2} (x) \sin (x) + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5

Simplify each term.

Tap for more steps...

Step 7.5.1

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

$6 x^{2} {(\frac{1}{\cos (x)})}^{2} \sin (x) + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5.2

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

$6 x^{2} \frac{1^{2}}{\cos^{2} (x)} \sin (x) + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5.3

One to any power is one.

$6 x^{2} \frac{1}{\cos^{2} (x)} \sin (x) + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5.4

Multiply $6 x^{2} \frac{1}{\cos^{2} (x)}$ .

Tap for more steps...

Step 7.5.4.1

Combine $6$ and $\frac{1}{\cos^{2} (x)}$ .

$x^{2} \frac{6}{\cos^{2} (x)} \sin (x) + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5.4.2

Combine $x^{2}$ and $\frac{6}{\cos^{2} (x)}$ .

$\frac{x^{2} \cdot 6}{\cos^{2} (x)} \sin (x) + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5.5

Move $6$ to the left of $x^{2}$ .

$\frac{6 \cdot x^{2}}{\cos^{2} (x)} \sin (x) + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5.6

Combine $\frac{6 x^{2}}{\cos^{2} (x)}$ and $\sin (x)$ .

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \cos (x) \tan (x) + 12 x \sin (x) \tan (x)$

Step 7.5.7

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

Tap for more steps...

Step 7.5.7.1

Add parentheses.

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} (\cos (x) \tan (x)) + 12 x \sin (x) \tan (x)$

Step 7.5.7.2

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

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} (\tan (x) \cos (x)) + 12 x \sin (x) \tan (x)$

Step 7.5.7.3

Rewrite $6 x^{2} \cos (x) \tan (x)$ in terms of sines and cosines.

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} (\frac{\sin (x)}{\cos (x)} \cos (x)) + 12 x \sin (x) \tan (x)$

Step 7.5.7.4

Cancel the common factors.

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + 12 x \sin (x) \tan (x)$

Step 7.5.8

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

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + 12 x \sin (x) \frac{\sin (x)}{\cos (x)}$

Step 7.5.9

Multiply $12 x \sin (x) \frac{\sin (x)}{\cos (x)}$ .

Tap for more steps...

Step 7.5.9.1

Combine $12$ and $\frac{\sin (x)}{\cos (x)}$ .

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + x \sin (x) \frac{12 \sin (x)}{\cos (x)}$

Step 7.5.9.2

Combine $\frac{12 \sin (x)}{\cos (x)}$ and $x$ .

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + \frac{12 \sin (x) x}{\cos (x)} \sin (x)$

Step 7.5.9.3

Combine $\frac{12 \sin (x) x}{\cos (x)}$ and $\sin (x)$ .

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + \frac{12 \sin (x) x \sin (x)}{\cos (x)}$

Step 7.5.9.4

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

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + \frac{12 x (\sin^{1} (x) \sin (x))}{\cos (x)}$

Step 7.5.9.5

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

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + \frac{12 x (\sin^{1} (x) \sin^{1} (x))}{\cos (x)}$

Step 7.5.9.6

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

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + \frac{12 x {\sin (x)}^{1 + 1}}{\cos (x)}$

Step 7.5.9.7

Add $1$ and $1$ .

$\frac{6 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6

Simplify each term.

Tap for more steps...

Step 7.6.1

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

$\frac{6 x^{2} \sin (x)}{\cos (x) \cos (x)} + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.2

Separate fractions.

$\frac{6 x^{2}}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.3

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

$\frac{6 x^{2}}{\cos (x)} \tan (x) + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.4

Combine $\frac{6 x^{2}}{\cos (x)}$ and $\tan (x)$ .

$\frac{6 x^{2} \tan (x)}{\cos (x)} + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.5

Separate fractions.

$\frac{6 x^{2}}{1} \cdot \frac{\tan (x)}{\cos (x)} + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.6

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

$\frac{6 x^{2}}{1} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.7

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$\frac{6 x^{2}}{1} (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.8

Simplify.

Tap for more steps...

Step 7.6.8.1

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

$\frac{6 x^{2}}{1} (\tan (x) \frac{1}{\cos (x)}) + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.8.2

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

$\frac{6 x^{2}}{1} (\tan (x) \sec (x)) + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.9

Divide $6 x^{2}$ by $1$ .

$6 x^{2} (\tan (x) \sec (x)) + 6 x^{2} \sin (x) + \frac{12 x \sin^{2} (x)}{\cos (x)}$

Step 7.6.10

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

$6 x^{2} (\tan (x) \sec (x)) + 6 x^{2} \sin (x) + \frac{12 x (\sin (x) \sin (x))}{\cos (x)}$

Step 7.6.11

Separate fractions.

$6 x^{2} (\tan (x) \sec (x)) + 6 x^{2} \sin (x) + \frac{12 x (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 7.6.12

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

$6 x^{2} (\tan (x) \sec (x)) + 6 x^{2} \sin (x) + \frac{12 x (\sin (x))}{1} \tan (x)$

Step 7.6.13

Divide $12 x (\sin (x))$ by $1$ .

$6 x^{2} (\tan (x) \sec (x)) + 6 x^{2} \sin (x) + 12 x (\sin (x)) \tan (x)$

Step 7.7

Reorder factors in $6 x^{2} (\tan (x) \sec (x)) + 6 x^{2} \sin (x) + 12 x (\sin (x)) \tan (x)$ .

$6 x^{2} \tan (x) \sec (x) + 6 x^{2} \sin (x) + 12 x (\sin (x)) \tan (x)$