Calculus Examples

$y = \ln (\sec (x) + \tan (x))$

Step 1

Differentiate both sides of the equation.

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

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 chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \sec (x) + \tan (x)$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $\sec (x) + \tan (x)$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sec (x) + \tan (x)]$

Step 3.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 3.1.3

Replace all occurrences of $u$ with $\sec (x) + \tan (x)$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify.

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Step 3.5.1

Reorder the factors of $\frac{1}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \sec^{2} (x))$ .

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

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

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Step 3.5.3.1

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

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

Step 3.5.3.2

Combine $\frac{\sec (x)}{\sec (x) + \tan (x)}$ and $\tan (x)$ .

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

Step 3.5.4

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

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

Step 3.5.5

Combine the numerators over the common denominator.

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

Step 3.5.6

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

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Step 3.5.6.1

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

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

Step 3.5.6.2

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

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

Step 3.5.6.3

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

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

Step 4

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

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

Step 5

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

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