Calculus Examples

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

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

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify the numerator.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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Step 6.2.1.2.1

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

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

Step 6.2.1.2.2

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

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

Step 6.2.1.2.3

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

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

Step 6.2.1.2.4

Add $1$ and $1$ .

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

Step 6.2.1.3

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

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Step 6.2.1.3.1

Move $\sec^{2} (x)$ .

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

Step 6.2.1.3.2

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

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Step 6.2.1.3.2.1

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

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

Step 6.2.1.3.2.2

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

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

Step 6.2.1.3.3

Add $2$ and $1$ .

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

Step 6.2.2

Factor $\sec (x)$ out of $\sec (x) \tan (x) + \tan^{2} (x) \sec (x) - \sec^{3} (x)$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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

Step 6.2.2.3

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

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

Step 6.2.2.4

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

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

Step 6.2.2.5

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

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

Step 6.2.3

Reorder $\tan^{2} (x)$ and $- \sec^{2} (x)$ .

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

Step 6.2.4

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

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

Step 6.2.5

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

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

Step 6.2.6

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

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

Step 6.2.7

Apply pythagorean identity.

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

Step 6.2.8

Multiply $- 1$ by $1$ .

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

Step 6.2.9

Apply the distributive property.

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

Step 6.2.10

Move $- 1$ to the left of $\sec (x)$ .

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

Step 6.2.11

Rewrite $- 1 \sec (x)$ as $- \sec (x)$ .

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

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

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