Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = \cos (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\sec (x) (- \sin (x))$

Step 3.4

Simplify.

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Step 3.4.1

Reorder the factors of $\sec (x) (- \sin (x))$ .

$- \sec (x) \sin (x)$

Step 3.4.2

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

$- \frac{1}{\cos (x)} \sin (x)$

Step 3.4.3

Combine $\sin (x)$ and $\frac{1}{\cos (x)}$ .

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

Step 3.4.4

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

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