Calculus Examples

$x^{2} \sin (x) \tan (x)$

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Remove unnecessary parentheses.

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

Step 6.3

Reorder terms.

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

Step 6.4

Simplify each term.

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

One to any power is one.

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

Step 6.4.4

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

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

Step 6.4.5

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

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

Step 6.4.6

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

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Step 6.4.6.1

Add parentheses.

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

Step 6.4.6.2

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

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

Step 6.4.6.3

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

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

Step 6.4.6.4

Cancel the common factors.

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

Step 6.4.7

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

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

Step 6.4.8

Multiply $2 x \sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Step 6.4.8.1

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

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

Step 6.4.8.2

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

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

Step 6.4.8.3

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

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

Step 6.4.8.4

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

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

Step 6.4.8.5

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

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

Step 6.4.8.6

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

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

Step 6.4.8.7

Add $1$ and $1$ .

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

Step 6.5

Simplify each term.

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Step 6.5.1

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

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

Step 6.5.2

Separate fractions.

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

Step 6.5.3

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

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

Step 6.5.4

Separate fractions.

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

Step 6.5.5

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

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

Step 6.5.6

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

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

Step 6.5.7

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

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

Step 6.5.8

Separate fractions.

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

Step 6.5.9

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

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

Step 6.5.10

Divide $2 x (\sin (x))$ by $1$ .

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