Calculus Examples

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

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 $f (x) = \ln (x)$ and $g (x) = \sin (x)$ .

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

Step 2

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

$\frac{\sin (x) \frac{1}{x} - \ln (x) \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}$

Step 3

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

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

Combine.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Cancel the common factor of $x$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite using the commutative property of multiplication.

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

Step 7.2

Reorder terms.

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Rewrite $- (x \cos (x) \ln (x) - \sin (x))$ as $- 1 (x \cos (x) \ln (x) - \sin (x))$ .

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

Step 7.7

Move the negative in front of the fraction.

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