Calculus Examples

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

Step 1

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

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

Step 2

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 $g (x) = x$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 4.2

Multiply $- 1$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Rewrite using the commutative property of multiplication.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Move the negative in front of the fraction.

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