Calculus Examples

$f (x) = \frac{\cos (x)}{x^{3}}$

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 $g (x) = x^{3}$ .

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

Step 2

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

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.1.2

Multiply $3$ by $- 1$ .

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

Step 4.1.2

Reorder factors in $- x^{3} \sin (x) - 3 \cos (x) x^{2}$ .

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

Step 4.2

Combine terms.

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Step 4.2.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

Multiply $3$ by $2$ .

$\frac{- x^{3} \sin (x) - 3 x^{2} \cos (x)}{x^{6}}$

Step 4.2.2

Factor $x^{2}$ out of $- x^{3} \sin (x) - 3 x^{2} \cos (x)$ .

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Step 4.2.2.1

Factor $x^{2}$ out of $- x^{3} \sin (x)$ .

$\frac{x^{2} (- x \sin (x)) - 3 x^{2} \cos (x)}{x^{6}}$

Step 4.2.2.2

Factor $x^{2}$ out of $- 3 x^{2} \cos (x)$ .

$\frac{x^{2} (- x \sin (x)) + x^{2} (- 3 \cos (x))}{x^{6}}$

Step 4.2.2.3

Factor $x^{2}$ out of $x^{2} (- x \sin (x)) + x^{2} (- 3 \cos (x))$ .

$\frac{x^{2} (- x \sin (x) - 3 \cos (x))}{x^{6}}$

Step 4.2.3

Cancel the common factors.

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Step 4.2.3.1

Factor $x^{2}$ out of $x^{6}$ .

$\frac{x^{2} (- x \sin (x) - 3 \cos (x))}{x^{2} x^{4}}$

Step 4.2.3.2

Cancel the common factor.

$\frac{x^{2} (- x \sin (x) - 3 \cos (x))}{x^{2} x^{4}}$

Step 4.2.3.3

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

Factor $- 1$ out of $- 3 \cos (x)$ .

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

Step 4.5

Factor $- 1$ out of $- (x \sin (x)) - (3 \cos (x))$ .

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

Step 4.6

Rewrite $- (x \sin (x) + 3 \cos (x))$ as $- 1 (x \sin (x) + 3 \cos (x))$ .

$\frac{- 1 (x \sin (x) + 3 \cos (x))}{x^{4}}$

Step 4.7

Move the negative in front of the fraction.

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