Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 2$ .

$\cos (x) (- 2 x^{- 3}) + x^{- 2} \frac{d}{d x} [\cos (x)]$

Step 3

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

$\cos (x) (- 2 x^{- 3}) + x^{- 2} (- \sin (x))$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Reorder terms.

$- 2 x^{- 3} \cos (x) - x^{- 2} \sin (x)$

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 \frac{1}{x^{3}} \cos (x) - x^{- 2} \sin (x)$

Step 4.2.2

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$\frac{- 2}{x^{3}} \cos (x) - x^{- 2} \sin (x)$

Step 4.2.3

Move the negative in front of the fraction.

$- \frac{2}{x^{3}} \cos (x) - x^{- 2} \sin (x)$

Step 4.2.4

Combine $\cos (x)$ and $\frac{2}{x^{3}}$ .

$- \frac{\cos (x) \cdot 2}{x^{3}} - x^{- 2} \sin (x)$

Step 4.2.5

Move $2$ to the left of $\cos (x)$ .

$- \frac{2 \cdot \cos (x)}{x^{3}} - x^{- 2} \sin (x)$

Step 4.2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2 \cos (x)}{x^{3}} - \frac{1}{x^{2}} \sin (x)$

Step 4.2.7

Combine $\sin (x)$ and $\frac{1}{x^{2}}$ .

$- \frac{2 \cos (x)}{x^{3}} - \frac{\sin (x)}{x^{2}}$