Calculus Examples

$(1 + \sec (x)) \sin (x)$

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 $f (x) = 1 + \sec (x)$ and $g (x) = \sin (x)$ .

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

Step 2

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

$(1 + \sec (x)) \cos (x) + \sin (x) \frac{d}{d x} [1 + \sec (x)]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 + \sec (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sec (x)]$ .

$(1 + \sec (x)) \cos (x) + \sin (x) (\frac{d}{d x} [1] + \frac{d}{d x} [\sec (x)])$

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(1 + \sec (x)) \cos (x) + \sin (x) (0 + \frac{d}{d x} [\sec (x)])$

Step 3.3

Add $0$ and $\frac{d}{d x} [\sec (x)]$ .

$(1 + \sec (x)) \cos (x) + \sin (x) \frac{d}{d x} [\sec (x)]$

Step 4

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

$(1 + \sec (x)) \cos (x) + \sin (x) (\sec (x) \tan (x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$1 \cos (x) + \sec (x) \cos (x) + \sin (x) \sec (x) \tan (x)$

Step 5.2

Multiply $\cos (x)$ by $1$ .

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

Step 5.3

Reorder terms.

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

Step 5.4

Simplify each term.

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Step 5.4.1

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

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Step 5.4.1.1

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

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

Step 5.4.1.2

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

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

Step 5.4.1.3

Cancel the common factors.

$1 + \sec (x) \sin (x) \tan (x) + \cos (x)$

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

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

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

Step 5.4.5

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

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Step 5.4.5.1

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

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

Step 5.4.5.2

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

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

Step 5.4.5.3

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

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

Step 5.4.5.4

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

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

Step 5.4.5.5

Add $1$ and $1$ .

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

Step 5.4.5.6

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

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

Step 5.4.5.7

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

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

Step 5.4.5.8

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

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

Step 5.4.5.9

Add $1$ and $1$ .

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

Step 5.5

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

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

Step 5.6

Rearrange terms.

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

Step 5.7

Apply pythagorean identity.

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