Calculus Examples

$g (x) = (\frac{5 x}{{(x^{3} - 6)}^{2}})$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5 x}{{(x^{3} - 6)}^{2}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{x}{{(x^{3} - 6)}^{2}}]$ .

$5 \frac{d}{d x} [\frac{x}{{(x^{3} - 6)}^{2}}]$

Step 2

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 $f (x) = x$ and $g (x) = {(x^{3} - 6)}^{2}$ .

$5 \frac{{(x^{3} - 6)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{3} - 6)}^{2}]}{{({(x^{3} - 6)}^{2})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(x^{3} - 6)}^{2})}^{2}$ .

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Step 3.1.1

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

$5 \frac{{(x^{3} - 6)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{3} - 6)}^{2}]}{{(x^{3} - 6)}^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$5 \frac{{(x^{3} - 6)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{3} - 6)}^{2}]}{{(x^{3} - 6)}^{4}}$

Step 3.2

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

$5 \frac{{(x^{3} - 6)}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{3} - 6)}^{2}]}{{(x^{3} - 6)}^{4}}$

Step 3.3

Multiply ${(x^{3} - 6)}^{2}$ by $1$ .

$5 \frac{{(x^{3} - 6)}^{2} - x \frac{d}{d x} [{(x^{3} - 6)}^{2}]}{{(x^{3} - 6)}^{4}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{3} - 6$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x^{3} - 6$ .

$5 \frac{{(x^{3} - 6)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{3} - 6])}{{(x^{3} - 6)}^{4}}$

Step 4.2

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

$5 \frac{{(x^{3} - 6)}^{2} - x (2 u \frac{d}{d x} [x^{3} - 6])}{{(x^{3} - 6)}^{4}}$

Step 4.3

Replace all occurrences of $u$ with $x^{3} - 6$ .

$5 \frac{{(x^{3} - 6)}^{2} - x (2 (x^{3} - 6) \frac{d}{d x} [x^{3} - 6])}{{(x^{3} - 6)}^{4}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $2$ by $- 1$ .

$5 \frac{{(x^{3} - 6)}^{2} - 2 x ((x^{3} - 6) \frac{d}{d x} [x^{3} - 6])}{{(x^{3} - 6)}^{4}}$

Step 5.2

Factor $x^{3} - 6$ out of ${(x^{3} - 6)}^{2} - 2 x ((x^{3} - 6) \frac{d}{d x} [x^{3} - 6])$ .

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Step 5.2.1

Factor $x^{3} - 6$ out of ${(x^{3} - 6)}^{2}$ .

$5 \frac{(x^{3} - 6) (x^{3} - 6) - 2 x ((x^{3} - 6) \frac{d}{d x} [x^{3} - 6])}{{(x^{3} - 6)}^{4}}$

Step 5.2.2

Factor $x^{3} - 6$ out of $- 2 x ((x^{3} - 6) \frac{d}{d x} [x^{3} - 6])$ .

$5 \frac{(x^{3} - 6) (x^{3} - 6) + (x^{3} - 6) (- 2 x (\frac{d}{d x} [x^{3} - 6]))}{{(x^{3} - 6)}^{4}}$

Step 5.2.3

Factor $x^{3} - 6$ out of $(x^{3} - 6) (x^{3} - 6) + (x^{3} - 6) (- 2 x (\frac{d}{d x} [x^{3} - 6]))$ .

$5 \frac{(x^{3} - 6) (x^{3} - 6 - 2 x (\frac{d}{d x} [x^{3} - 6]))}{{(x^{3} - 6)}^{4}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $x^{3} - 6$ out of ${(x^{3} - 6)}^{4}$ .

$5 \frac{(x^{3} - 6) (x^{3} - 6 - 2 x (\frac{d}{d x} [x^{3} - 6]))}{(x^{3} - 6) {(x^{3} - 6)}^{3}}$

Step 6.2

Cancel the common factor.

$5 \frac{(x^{3} - 6) (x^{3} - 6 - 2 x (\frac{d}{d x} [x^{3} - 6]))}{(x^{3} - 6) {(x^{3} - 6)}^{3}}$

Step 6.3

Rewrite the expression.

$5 \frac{x^{3} - 6 - 2 x (\frac{d}{d x} [x^{3} - 6])}{{(x^{3} - 6)}^{3}}$

Step 7

By the Sum Rule, the derivative of $x^{3} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6]$ .

$5 \frac{x^{3} - 6 - 2 x (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6])}{{(x^{3} - 6)}^{3}}$

Step 8

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

$5 \frac{x^{3} - 6 - 2 x (3 x^{2} + \frac{d}{d x} [- 6])}{{(x^{3} - 6)}^{3}}$

Step 9

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$5 \frac{x^{3} - 6 - 2 x (3 x^{2} + 0)}{{(x^{3} - 6)}^{3}}$

Step 10

Simplify the expression.

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Step 10.1

Add $3 x^{2}$ and $0$ .

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

Step 10.2

Multiply $3$ by $- 2$ .

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

Step 11

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 11.1

Move $x^{2}$ .

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

Step 11.2

Multiply $x^{2}$ by $x$ .

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Step 11.2.1

Raise $x$ to the power of $1$ .

$5 \frac{x^{3} - 6 - 6 (x^{2} x^{1})}{{(x^{3} - 6)}^{3}}$

Step 11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 \frac{x^{3} - 6 - 6 x^{2 + 1}}{{(x^{3} - 6)}^{3}}$

Step 11.3

Add $2$ and $1$ .

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

Step 12

Subtract $6 x^{3}$ from $x^{3}$ .

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

Step 13

Combine $5$ and $\frac{- 5 x^{3} - 6}{{(x^{3} - 6)}^{3}}$ .

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

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{5 (- 5 x^{3}) + 5 \cdot - 6}{{(x^{3} - 6)}^{3}}$

Step 14.2

Simplify each term.

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Step 14.2.1

Multiply $- 5$ by $5$ .

$\frac{- 25 x^{3} + 5 \cdot - 6}{{(x^{3} - 6)}^{3}}$

Step 14.2.2

Multiply $5$ by $- 6$ .

$\frac{- 25 x^{3} - 30}{{(x^{3} - 6)}^{3}}$

Step 14.3

Factor $5$ out of $- 25 x^{3} - 30$ .

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Step 14.3.1

Factor $5$ out of $- 25 x^{3}$ .

$\frac{5 (- 5 x^{3}) - 30}{{(x^{3} - 6)}^{3}}$

Step 14.3.2

Factor $5$ out of $- 30$ .

$\frac{5 (- 5 x^{3}) + 5 (- 6)}{{(x^{3} - 6)}^{3}}$

Step 14.3.3

Factor $5$ out of $5 (- 5 x^{3}) + 5 (- 6)$ .

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

Step 14.4

Factor $- 1$ out of $- 5 x^{3}$ .

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

Step 14.5

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{5 (- (5 x^{3}) - 1 (6))}{{(x^{3} - 6)}^{3}}$

Step 14.6

Factor $- 1$ out of $- (5 x^{3}) - 1 (6)$ .

$\frac{5 (- (5 x^{3} + 6))}{{(x^{3} - 6)}^{3}}$

Step 14.7

Rewrite $- (5 x^{3} + 6)$ as $- 1 (5 x^{3} + 6)$ .

$\frac{5 (- 1 (5 x^{3} + 6))}{{(x^{3} - 6)}^{3}}$

Step 14.8

Move the negative in front of the fraction.

$- \frac{5 (5 x^{3} + 6)}{{(x^{3} - 6)}^{3}}$