Calculus Examples

$f (x) = \sin (x) \sec^{2} (x)$

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 3

Move $2$ to the left of $\sin (x)$ .

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

Step 4

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Reorder terms.

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

Step 10.2

Simplify each term.

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.2.3

One to any power is one.

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

Step 10.2.4

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

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

Step 10.2.5

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

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

Step 10.2.6

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

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

Step 10.2.7

Combine.

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

Step 10.2.8

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

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Step 10.2.8.1

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 10.2.8.1.1

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

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

Step 10.2.8.1.2

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

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

Step 10.2.8.2

Add $2$ and $1$ .

$\frac{2 \sin (x) \sin (x)}{\cos^{3} (x)} + \sec^{2} (x) \cos (x)$

Step 10.2.9

Simplify the numerator.

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Step 10.2.9.1

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

$\frac{2 (\sin^{1} (x) \sin (x))}{\cos^{3} (x)} + \sec^{2} (x) \cos (x)$

Step 10.2.9.2

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

$\frac{2 (\sin^{1} (x) \sin^{1} (x))}{\cos^{3} (x)} + \sec^{2} (x) \cos (x)$

Step 10.2.9.3

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

$\frac{2 {\sin (x)}^{1 + 1}}{\cos^{3} (x)} + \sec^{2} (x) \cos (x)$

Step 10.2.9.4

Add $1$ and $1$ .

$\frac{2 \sin^{2} (x)}{\cos^{3} (x)} + \sec^{2} (x) \cos (x)$

Step 10.2.10

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

$\frac{2 \sin^{2} (x)}{\cos^{3} (x)} + {(\frac{1}{\cos (x)})}^{2} \cos (x)$

Step 10.2.11

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

$\frac{2 \sin^{2} (x)}{\cos^{3} (x)} + \frac{1^{2}}{\cos^{2} (x)} \cos (x)$

Step 10.2.12

Cancel the common factor of $\cos (x)$ .

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Step 10.2.12.1

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

$\frac{2 \sin^{2} (x)}{\cos^{3} (x)} + \frac{1^{2}}{\cos (x) \cos (x)} \cos (x)$

Step 10.2.12.2

Cancel the common factor.

$\frac{2 \sin^{2} (x)}{\cos^{3} (x)} + \frac{1^{2}}{\cos (x) \cos (x)} \cos (x)$

Step 10.2.12.3

Rewrite the expression.

$\frac{2 \sin^{2} (x)}{\cos^{3} (x)} + \frac{1^{2}}{\cos (x)}$

Step 10.2.13

One to any power is one.

$\frac{2 \sin^{2} (x)}{\cos^{3} (x)} + \frac{1}{\cos (x)}$

Step 10.3

Simplify each term.

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Step 10.3.1

Multiply by $1$ .

$\frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{3} (x)} + \frac{1}{\cos (x)}$

Step 10.3.2

Factor $\cos^{2} (x)$ out of $\cos^{3} (x)$ .

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

Step 10.3.3

Separate fractions.

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

Step 10.3.4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 10.3.5

Multiply $2$ by $1$ .

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

Step 10.3.6

Separate fractions.

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

Step 10.3.7

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

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

Step 10.3.8

Divide $2$ by $1$ .

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

Step 10.3.9

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

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