Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 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 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 3.4

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

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

Step 3.5

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

$\cos (x) (\sec (x) \tan (x))$

Step 3.6

Simplify.

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Step 3.6.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.6.1.1

Reorder $\cos (x)$ and $\sec (x)$ .

$\sec (x) \cos (x) \tan (x)$

Step 3.6.1.2

Rewrite $\cos (x) \sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} \cos (x) \tan (x)$

Step 3.6.1.3

Cancel the common factors.

$1 \tan (x)$

Step 3.6.2

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

$\tan (x)$

Step 3.6.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)}$

Step 3.6.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x)$

Step 4

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

$y' = \tan (x)$

Step 5

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

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