Calculus Examples

$\frac{\tan (x) - 1}{\sec (x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \tan (x) - 1$ and $g (x) = \sec (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$\frac{\sec (x) (\sec^{2} (x) + 0) - (\tan (x) - 1) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Step 4.2

Add $\sec^{2} (x)$ and $0$ .

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Add $1$ and $2$ .

$\frac{\sec^{3} (x) - (\tan (x) - 1) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Step 6

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

$\frac{\sec^{3} (x) - (\tan (x) - 1) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Step 7

Factor $\sec (x)$ out of $\sec^{3} (x) - (\tan (x) - 1) \sec (x) \tan (x)$ .

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Step 7.1

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

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

Step 7.2

Factor $\sec (x)$ out of $- (\tan (x) - 1) \sec (x) \tan (x)$ .

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

Step 7.3

Factor $\sec (x)$ out of $\sec (x) \sec^{2} (x) + \sec (x) (- (\tan (x) - 1) \tan (x))$ .

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

Step 8

Cancel the common factors.

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Step 8.1

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

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Simplify the numerator.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

Multiply $- \tan (x) \tan (x)$ .

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Step 9.3.1.1.1

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

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

Step 9.3.1.1.2

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

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

Step 9.3.1.1.3

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

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

Step 9.3.1.1.4

Add $1$ and $1$ .

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

Step 9.3.1.2

Multiply $- 1$ by $- 1$ .

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

Step 9.3.1.3

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

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

Step 9.3.2

Apply pythagorean identity.

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