Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $3$ by $2$ .

$\frac{x^{3} \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.3

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

$\frac{x^{3} (- \sin (x)) - \cos (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

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

Step 3.4.2

Simplify with factoring out.

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Step 3.4.2.1

Multiply $3$ by $- 1$ .

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

Step 3.4.2.2

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

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

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

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

Step 3.4.2.2.3

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 3.5

Cancel the common factors.

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Step 3.5.1

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

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

Step 3.5.2

Cancel the common factor.

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

Step 3.5.3

Rewrite the expression.

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

Step 3.6

Simplify.

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Step 3.6.1

Rewrite using the commutative property of multiplication.

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

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 3.6.6

Move the negative in front of the fraction.

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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