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

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

Step 2

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

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

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

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

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Step 9.2.1.1

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

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

Step 9.2.1.2

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

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

Step 9.2.1.3

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

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

Step 9.2.1.4

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

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

Step 9.2.1.5

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

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

Step 9.2.2

Apply pythagorean identity.

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

Step 9.2.3

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

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

Step 9.2.4

Apply the distributive property.

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

Step 9.2.5

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

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Step 9.2.5.1

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

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

Step 9.2.5.2

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

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

Step 9.2.5.3

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

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

Step 9.2.5.4

Add $1$ and $1$ .

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

Step 9.2.6

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

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

Multiply by $1$ .

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

Step 9.3.3

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

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

Step 9.4

Cancel the common factor of $\sec (x) + 1$ and ${(1 + \sec (x))}^{2}$ .

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Step 9.4.1

Reorder terms.

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

Step 9.4.2

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

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

Step 9.4.3

Cancel the common factors.

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Step 9.4.3.1

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

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

Step 9.4.3.2

Cancel the common factor.

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

Step 9.4.3.3

Rewrite the expression.

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