Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \cos (x)$ and $g (x) = \ln (x)$ .

$\frac{\ln (x) \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}$

Step 2

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

$\frac{\ln (x) (- \sin (x)) - \cos (x) \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}$

Step 3

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

$\frac{\ln (x) (- \sin (x)) - \cos (x) \frac{1}{x}}{\ln^{2} (x)}$

Step 4

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

$\frac{\ln (x) (- \sin (x)) - \frac{\cos (x)}{x}}{\ln^{2} (x)}$

Step 5

Simplify.

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Step 5.1

Rewrite using the commutative property of multiplication.

$\frac{- \ln (x) \sin (x) - \frac{\cos (x)}{x}}{\ln^{2} (x)}$

Step 5.2

Reorder terms.

$\frac{- \sin (x) \ln (x) - \frac{\cos (x)}{x}}{\ln^{2} (x)}$

Step 5.3

Factor $- 1$ out of $- \sin (x) \ln (x) - \frac{\cos (x)}{x}$ .

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Step 5.3.1

Factor $- 1$ out of $- \sin (x) \ln (x)$ .

$\frac{- (\sin (x) \ln (x)) - \frac{\cos (x)}{x}}{\ln^{2} (x)}$

Step 5.3.2

Factor $- 1$ out of $- \frac{\cos (x)}{x}$ .

$\frac{- (\sin (x) \ln (x)) - (\frac{\cos (x)}{x})}{\ln^{2} (x)}$

Step 5.3.3

Factor $- 1$ out of $- (\sin (x) \ln (x)) - (\frac{\cos (x)}{x})$ .

$\frac{- (\sin (x) \ln (x) + \frac{\cos (x)}{x})}{\ln^{2} (x)}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{\sin (x) \ln (x) + \frac{\cos (x)}{x}}{\ln^{2} (x)}$