Calculus Examples

$\frac{x}{{(9 x - 4)}^{3}}$

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 $f (x) = x$ and $g (x) = {(9 x - 4)}^{3}$ .

$\frac{{(9 x - 4)}^{3} \frac{d}{d x} [x] - x \frac{d}{d x} [{(9 x - 4)}^{3}]}{{({(9 x - 4)}^{3})}^{2}}$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Multiply the exponents in ${({(9 x - 4)}^{3})}^{2}$ .

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Step 2.1.1

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

$\frac{{(9 x - 4)}^{3} \frac{d}{d x} [x] - x \frac{d}{d x} [{(9 x - 4)}^{3}]}{{(9 x - 4)}^{3 \cdot 2}}$

Step 2.1.2

Multiply $3$ by $2$ .

$\frac{{(9 x - 4)}^{3} \frac{d}{d x} [x] - x \frac{d}{d x} [{(9 x - 4)}^{3}]}{{(9 x - 4)}^{6}}$

Step 2.2

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

$\frac{{(9 x - 4)}^{3} \cdot 1 - x \frac{d}{d x} [{(9 x - 4)}^{3}]}{{(9 x - 4)}^{6}}$

Step 2.3

Multiply ${(9 x - 4)}^{3}$ by $1$ .

$\frac{{(9 x - 4)}^{3} - x \frac{d}{d x} [{(9 x - 4)}^{3}]}{{(9 x - 4)}^{6}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 9 x - 4$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $9 x - 4$ .

$\frac{{(9 x - 4)}^{3} - x (\frac{d}{d u} [u^{3}] \frac{d}{d x} [9 x - 4])}{{(9 x - 4)}^{6}}$

Step 3.2

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

$\frac{{(9 x - 4)}^{3} - x (3 u^{2} \frac{d}{d x} [9 x - 4])}{{(9 x - 4)}^{6}}$

Step 3.3

Replace all occurrences of $u$ with $9 x - 4$ .

$\frac{{(9 x - 4)}^{3} - x (3 {(9 x - 4)}^{2} \frac{d}{d x} [9 x - 4])}{{(9 x - 4)}^{6}}$

Step 4

Simplify with factoring out.

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Step 4.1

Multiply $3$ by $- 1$ .

$\frac{{(9 x - 4)}^{3} - 3 x ({(9 x - 4)}^{2} \frac{d}{d x} [9 x - 4])}{{(9 x - 4)}^{6}}$

Step 4.2

Factor ${(9 x - 4)}^{2}$ out of ${(9 x - 4)}^{3} - 3 x ({(9 x - 4)}^{2} \frac{d}{d x} [9 x - 4])$ .

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Step 4.2.1

Factor ${(9 x - 4)}^{2}$ out of ${(9 x - 4)}^{3}$ .

$\frac{{(9 x - 4)}^{2} (9 x - 4) - 3 x ({(9 x - 4)}^{2} \frac{d}{d x} [9 x - 4])}{{(9 x - 4)}^{6}}$

Step 4.2.2

Factor ${(9 x - 4)}^{2}$ out of $- 3 x ({(9 x - 4)}^{2} \frac{d}{d x} [9 x - 4])$ .

$\frac{{(9 x - 4)}^{2} (9 x - 4) + {(9 x - 4)}^{2} (- 3 x (\frac{d}{d x} [9 x - 4]))}{{(9 x - 4)}^{6}}$

Step 4.2.3

Factor ${(9 x - 4)}^{2}$ out of ${(9 x - 4)}^{2} (9 x - 4) + {(9 x - 4)}^{2} (- 3 x (\frac{d}{d x} [9 x - 4]))$ .

$\frac{{(9 x - 4)}^{2} (9 x - 4 - 3 x (\frac{d}{d x} [9 x - 4]))}{{(9 x - 4)}^{6}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor ${(9 x - 4)}^{2}$ out of ${(9 x - 4)}^{6}$ .

$\frac{{(9 x - 4)}^{2} (9 x - 4 - 3 x (\frac{d}{d x} [9 x - 4]))}{{(9 x - 4)}^{2} {(9 x - 4)}^{4}}$

Step 5.2

Cancel the common factor.

$\frac{{(9 x - 4)}^{2} (9 x - 4 - 3 x (\frac{d}{d x} [9 x - 4]))}{{(9 x - 4)}^{2} {(9 x - 4)}^{4}}$

Step 5.3

Rewrite the expression.

$\frac{9 x - 4 - 3 x (\frac{d}{d x} [9 x - 4])}{{(9 x - 4)}^{4}}$

Step 6

By the Sum Rule, the derivative of $9 x - 4$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [- 4]$ .

$\frac{9 x - 4 - 3 x (\frac{d}{d x} [9 x] + \frac{d}{d x} [- 4])}{{(9 x - 4)}^{4}}$

Step 7

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$\frac{9 x - 4 - 3 x (9 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{{(9 x - 4)}^{4}}$

Step 8

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

$\frac{9 x - 4 - 3 x (9 \cdot 1 + \frac{d}{d x} [- 4])}{{(9 x - 4)}^{4}}$

Step 9

Multiply $9$ by $1$ .

$\frac{9 x - 4 - 3 x (9 + \frac{d}{d x} [- 4])}{{(9 x - 4)}^{4}}$

Step 10

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{9 x - 4 - 3 x (9 + 0)}{{(9 x - 4)}^{4}}$

Step 11

Simplify by adding terms.

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Step 11.1

Add $9$ and $0$ .

$\frac{9 x - 4 - 3 x \cdot 9}{{(9 x - 4)}^{4}}$

Step 11.2

Multiply $9$ by $- 3$ .

$\frac{9 x - 4 - 27 x}{{(9 x - 4)}^{4}}$

Step 11.3

Subtract $27 x$ from $9 x$ .

$\frac{- 18 x - 4}{{(9 x - 4)}^{4}}$