Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{\sec (x)}{1 + \sec (x)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = 1 + \sec (x)$ .

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Add $1$ and $1$ .

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

Step 3.9

Simplify.

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Step 3.9.1

Apply the distributive property.

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

Step 3.9.2

Apply the distributive property.

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

Step 3.9.3

Simplify the numerator.

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Step 3.9.3.1

Simplify each term.

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Step 3.9.3.1.1

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

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

Step 3.9.3.1.2

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

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Step 3.9.3.1.2.1

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

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

Step 3.9.3.1.2.2

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

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

Step 3.9.3.1.2.3

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

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

Step 3.9.3.1.2.4

Add $1$ and $1$ .

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

Step 3.9.3.2

Combine the opposite terms in $\sec (x) \tan (x) + \sec^{2} (x) \tan (x) - \sec^{2} (x) \tan (x)$ .

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Step 3.9.3.2.1

Subtract $\sec^{2} (x) \tan (x)$ from $\sec^{2} (x) \tan (x)$ .

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

Step 3.9.3.2.2

Add $\sec (x) \tan (x)$ and $0$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{\sec (x) \tan (x)}{{(1 + \sec (x))}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{\sec (x) \tan (x)}{{(1 + \sec (x))}^{2}}$