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 $u_{1}$ as $| \sec (x) + \tan (x) |$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $| \sec (x) + \tan (x) |$ .

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $\sec (x) + \tan (x)$ .

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

Step 3.2.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

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

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\sec (x) + \tan (x)$ .

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

Step 3.3

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

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

Step 3.4

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Differentiate using the Sum Rule.

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Step 3.8.1

Add $1$ and $1$ .

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

Step 3.8.2

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Simplify.

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Step 3.11.1

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

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

Step 3.11.2

Remove non-negative terms from the absolute value.

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

Step 3.11.3

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

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Step 3.11.3.1

Multiply by $1$ .

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

Step 3.11.3.2

Cancel the common factors.

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Step 3.11.3.2.1

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

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

Step 3.11.3.2.2

Cancel the common factor.

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

Step 3.11.3.2.3

Rewrite the expression.

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

Step 3.11.4

Apply the distributive property.

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

Step 3.11.5

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

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Step 3.11.5.1

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 3.11.5.2

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 3.11.6

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 3.11.7

Combine the numerators over the common denominator.

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

Step 3.11.8

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

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Step 3.11.8.1

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

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

Step 3.11.8.2

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

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

Step 3.11.8.3

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 $\frac{d y}{d x}$ .

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