Calculus Examples

$f (x) = 1 + 7 \sin (x) - \tan (x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $1 + 7 \sin (x) - \tan (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [7 \sin (x)] + \frac{d}{d x} [- \tan (x)]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [7 \sin (x)] + \frac{d}{d x} [- \tan (x)]$

Step 1.2

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

$0 + \frac{d}{d x} [7 \sin (x)] + \frac{d}{d x} [- \tan (x)]$

Step 2

Evaluate $\frac{d}{d x} [7 \sin (x)]$ .

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \sin (x)$ with respect to $x$ is $7 \frac{d}{d x} [\sin (x)]$ .

$0 + 7 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \tan (x)]$

Step 2.2

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

$0 + 7 \cos (x) + \frac{d}{d x} [- \tan (x)]$

Step 3

Evaluate $\frac{d}{d x} [- \tan (x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (x)$ with respect to $x$ is $- \frac{d}{d x} [\tan (x)]$ .

$0 + 7 \cos (x) - \frac{d}{d x} [\tan (x)]$

Step 3.2

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

$0 + 7 \cos (x) - \sec^{2} (x)$

Step 4

Simplify.

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Step 4.1

Add $0$ and $7 \cos (x)$ .

$7 \cos (x) - \sec^{2} (x)$

Step 4.2

Simplify each term.

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Step 4.2.1

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

$7 \cos (x) - {(\frac{1}{\cos (x)})}^{2}$

Step 4.2.2

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

$7 \cos (x) - \frac{1^{2}}{\cos^{2} (x)}$

Step 4.2.3

One to any power is one.

$7 \cos (x) - \frac{1}{\cos^{2} (x)}$

Step 4.3

Simplify each term.

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Step 4.3.1

Rewrite $1$ as $1^{2}$ .

$7 \cos (x) - \frac{1^{2}}{\cos^{2} (x)}$

Step 4.3.2

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$7 \cos (x) - {(\frac{1}{\cos (x)})}^{2}$

Step 4.3.3

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

$7 \cos (x) - \sec^{2} (x)$