Calculus Examples

$y = \ln (| \sec (x) |)$

Step 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 $g (x) = | \sec (x) |$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $| \sec (x) |$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 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 $g (x) = \sec (x)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

Combine $\sec (x)$ and $\frac{\sec (x)}{| \sec^{2} (x) |}$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Add $1$ and $1$ .

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

Step 15

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

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

Step 16

Simplify.

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Step 16.1

Remove non-negative terms from the absolute value.

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

Step 16.2

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

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Step 16.2.1

Cancel the common factor.

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

Step 16.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x)$