Calculus Examples

$y = \frac{3 \cos (x)}{5 x^{3}}$

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 $f (x) = 3 \cos (x)$ and $g (x) = 5 x^{3}$ .

$\frac{5 x^{3} \frac{d}{d x} [3 \cos (x)] - 1 \cdot 3 \cos (x) \frac{d}{d x} [5 x^{3}]}{{(5 x^{3})}^{2}}$

Step 2

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos (x)$ with respect to $x$ is $3 \frac{d}{d x} [\cos (x)]$ .

$\frac{5 x^{3} (3 \frac{d}{d x} [\cos (x)]) - 1 \cdot 3 \cos (x) \frac{d}{d x} [5 x^{3}]}{{(5 x^{3})}^{2}}$

Step 3

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

$\frac{5 x^{3} (3 (- \sin (x))) - 1 \cdot 3 \cos (x) \frac{d}{d x} [5 x^{3}]}{{(5 x^{3})}^{2}}$

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $3$ .

$\frac{5 x^{3} (- 3 \sin (x)) - 1 \cdot 3 \cos (x) \frac{d}{d x} [5 x^{3}]}{{(5 x^{3})}^{2}}$

Step 4.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$\frac{5 x^{3} (- 3 \sin (x)) - 1 \cdot 3 \cos (x) (5 \frac{d}{d x} [x^{3}])}{{(5 x^{3})}^{2}}$

Step 4.3

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

$\frac{5 x^{3} (- 3 \sin (x)) - 1 \cdot 3 \cos (x) (5 (3 x^{2}))}{{(5 x^{3})}^{2}}$

Step 4.4

Multiply $3$ by $5$ .

$\frac{5 x^{3} (- 3 \sin (x)) - 1 \cdot 3 \cos (x) (15 x^{2})}{{(5 x^{3})}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $5 x^{3}$ .

$\frac{5 x^{3} (- 3 \sin (x)) - 1 \cdot 3 \cos (x) (15 x^{2})}{5^{2} {(x^{3})}^{2}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $- 3$ by $5$ .

$\frac{- 15 x^{3} \sin (x) - 1 \cdot 3 \cos (x) (15 x^{2})}{5^{2} {(x^{3})}^{2}}$

Step 5.2.1.2

Multiply $- 1$ by $3$ .

$\frac{- 15 x^{3} \sin (x) - 3 \cos (x) (15 x^{2})}{5^{2} {(x^{3})}^{2}}$

Step 5.2.1.3

Multiply $15$ by $- 3$ .

$\frac{- 15 x^{3} \sin (x) - 45 \cos (x) x^{2}}{5^{2} {(x^{3})}^{2}}$

Step 5.2.2

Reorder factors in $- 15 x^{3} \sin (x) - 45 \cos (x) x^{2}$ .

$\frac{- 15 x^{3} \sin (x) - 45 x^{2} \cos (x)}{5^{2} {(x^{3})}^{2}}$

Step 5.3

Combine terms.

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Step 5.3.1

Raise $5$ to the power of $2$ .

$\frac{- 15 x^{3} \sin (x) - 45 x^{2} \cos (x)}{25 {(x^{3})}^{2}}$

Step 5.3.2

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

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Step 5.3.2.1

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

$\frac{- 15 x^{3} \sin (x) - 45 x^{2} \cos (x)}{25 x^{3 \cdot 2}}$

Step 5.3.2.2

Multiply $3$ by $2$ .

$\frac{- 15 x^{3} \sin (x) - 45 x^{2} \cos (x)}{25 x^{6}}$

Step 5.4

Factor $15 x^{2}$ out of $- 15 x^{3} \sin (x) - 45 x^{2} \cos (x)$ .

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Step 5.4.1

Factor $15 x^{2}$ out of $- 15 x^{3} \sin (x)$ .

$\frac{15 x^{2} (- x \sin (x)) - 45 x^{2} \cos (x)}{25 x^{6}}$

Step 5.4.2

Factor $15 x^{2}$ out of $- 45 x^{2} \cos (x)$ .

$\frac{15 x^{2} (- x \sin (x)) + 15 x^{2} (- 3 \cos (x))}{25 x^{6}}$

Step 5.4.3

Factor $15 x^{2}$ out of $15 x^{2} (- x \sin (x)) + 15 x^{2} (- 3 \cos (x))$ .

$\frac{15 x^{2} (- x \sin (x) - 3 \cos (x))}{25 x^{6}}$

Step 5.5

Cancel the common factor of $15$ and $25$ .

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Step 5.5.1

Factor $5$ out of $15 x^{2} (- x \sin (x) - 3 \cos (x))$ .

$\frac{5 (3 x^{2} (- x \sin (x) - 3 \cos (x)))}{25 x^{6}}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $5$ out of $25 x^{6}$ .

$\frac{5 (3 x^{2} (- x \sin (x) - 3 \cos (x)))}{5 (5 x^{6})}$

Step 5.5.2.2

Cancel the common factor.

$\frac{5 (3 x^{2} (- x \sin (x) - 3 \cos (x)))}{5 (5 x^{6})}$

Step 5.5.2.3

Rewrite the expression.

$\frac{3 x^{2} (- x \sin (x) - 3 \cos (x))}{5 x^{6}}$

Step 5.6

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 5.6.1

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

$\frac{x^{2} (3 (- x \sin (x) - 3 \cos (x)))}{5 x^{6}}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $x^{2}$ out of $5 x^{6}$ .

$\frac{x^{2} (3 (- x \sin (x) - 3 \cos (x)))}{x^{2} (5 x^{4})}$

Step 5.6.2.2

Cancel the common factor.

$\frac{x^{2} (3 (- x \sin (x) - 3 \cos (x)))}{x^{2} (5 x^{4})}$

Step 5.6.2.3

Rewrite the expression.

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

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

$\frac{3 (- 1 (x \sin (x) + 3 \cos (x)))}{5 x^{4}}$

Step 5.11

Move the negative in front of the fraction.

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