Calculus Examples

$f (x) = \frac{\cos (x)}{x^{4}}$

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 $g (x) = x^{4}$ .

$\frac{x^{4} \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2

Multiply the exponents in ${(x^{4})}^{2}$ .

Tap for more steps...

Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{4} \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [x^{4}]}{x^{4 \cdot 2}}$

Step 2.2

Multiply $4$ by $2$ .

$\frac{x^{4} \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 3

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

$\frac{x^{4} (- \sin (x)) - \cos (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4

Differentiate using the Power Rule.

Tap for more steps...

Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{x^{4} (- \sin (x)) - \cos (x) (4 x^{3})}{x^{8}}$

Step 4.2

Simplify with factoring out.

Tap for more steps...

Step 4.2.1

Multiply $4$ by $- 1$ .

$\frac{x^{4} (- \sin (x)) - 4 \cos (x) x^{3}}{x^{8}}$

Step 4.2.2

Factor $x^{3}$ out of $x^{4} (- \sin (x)) - 4 \cos (x) x^{3}$ .

Tap for more steps...

Step 4.2.2.1

Factor $x^{3}$ out of $x^{4} (- \sin (x))$ .

$\frac{x^{3} (x (- \sin (x))) - 4 \cos (x) x^{3}}{x^{8}}$

Step 4.2.2.2

Factor $x^{3}$ out of $- 4 \cos (x) x^{3}$ .

$\frac{x^{3} (x (- \sin (x))) + x^{3} (- 4 \cos (x))}{x^{8}}$

Step 4.2.2.3

Factor $x^{3}$ out of $x^{3} (x (- \sin (x))) + x^{3} (- 4 \cos (x))$ .

$\frac{x^{3} (x (- \sin (x)) - 4 \cos (x))}{x^{8}}$

Step 5

Cancel the common factors.

Tap for more steps...

Step 5.1

Factor $x^{3}$ out of $x^{8}$ .

$\frac{x^{3} (x (- \sin (x)) - 4 \cos (x))}{x^{3} x^{5}}$

Step 5.2

Cancel the common factor.

$\frac{x^{3} (x (- \sin (x)) - 4 \cos (x))}{x^{3} x^{5}}$

Step 5.3

Rewrite the expression.

$\frac{x (- \sin (x)) - 4 \cos (x)}{x^{5}}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Rewrite using the commutative property of multiplication.

$\frac{- x \sin (x) - 4 \cos (x)}{x^{5}}$

Step 6.2

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

$\frac{- (x \sin (x)) - 4 \cos (x)}{x^{5}}$

Step 6.3

Factor $- 1$ out of $- 4 \cos (x)$ .

$\frac{- (x \sin (x)) - (4 \cos (x))}{x^{5}}$

Step 6.4

Factor $- 1$ out of $- (x \sin (x)) - (4 \cos (x))$ .

$\frac{- (x \sin (x) + 4 \cos (x))}{x^{5}}$

Step 6.5

Rewrite $- (x \sin (x) + 4 \cos (x))$ as $- 1 (x \sin (x) + 4 \cos (x))$ .

$\frac{- 1 (x \sin (x) + 4 \cos (x))}{x^{5}}$

Step 6.6

Move the negative in front of the fraction.

$- \frac{x \sin (x) + 4 \cos (x)}{x^{5}}$