Calculus Examples

$\frac{x}{{(4 x - 11)}^{3}}$

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 $g (x) = {(4 x - 11)}^{3}$ .

$\frac{{(4 x - 11)}^{3} \frac{d}{d x} [x] - x \frac{d}{d x} [{(4 x - 11)}^{3}]}{{({(4 x - 11)}^{3})}^{2}}$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Multiply the exponents in ${({(4 x - 11)}^{3})}^{2}$ .

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Step 2.1.1

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

$\frac{{(4 x - 11)}^{3} \frac{d}{d x} [x] - x \frac{d}{d x} [{(4 x - 11)}^{3}]}{{(4 x - 11)}^{3 \cdot 2}}$

Step 2.1.2

Multiply $3$ by $2$ .

$\frac{{(4 x - 11)}^{3} \frac{d}{d x} [x] - x \frac{d}{d x} [{(4 x - 11)}^{3}]}{{(4 x - 11)}^{6}}$

Step 2.2

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

$\frac{{(4 x - 11)}^{3} \cdot 1 - x \frac{d}{d x} [{(4 x - 11)}^{3}]}{{(4 x - 11)}^{6}}$

Step 2.3

Multiply ${(4 x - 11)}^{3}$ by $1$ .

$\frac{{(4 x - 11)}^{3} - x \frac{d}{d x} [{(4 x - 11)}^{3}]}{{(4 x - 11)}^{6}}$

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 $g (x) = 4 x - 11$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $4 x - 11$ .

$\frac{{(4 x - 11)}^{3} - x (\frac{d}{d u} [u^{3}] \frac{d}{d x} [4 x - 11])}{{(4 x - 11)}^{6}}$

Step 3.2

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

$\frac{{(4 x - 11)}^{3} - x (3 u^{2} \frac{d}{d x} [4 x - 11])}{{(4 x - 11)}^{6}}$

Step 3.3

Replace all occurrences of $u$ with $4 x - 11$ .

$\frac{{(4 x - 11)}^{3} - x (3 {(4 x - 11)}^{2} \frac{d}{d x} [4 x - 11])}{{(4 x - 11)}^{6}}$

Step 4

Simplify with factoring out.

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Step 4.1

Multiply $3$ by $- 1$ .

$\frac{{(4 x - 11)}^{3} - 3 x ({(4 x - 11)}^{2} \frac{d}{d x} [4 x - 11])}{{(4 x - 11)}^{6}}$

Step 4.2

Factor ${(4 x - 11)}^{2}$ out of ${(4 x - 11)}^{3} - 3 x ({(4 x - 11)}^{2} \frac{d}{d x} [4 x - 11])$ .

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Step 4.2.1

Factor ${(4 x - 11)}^{2}$ out of ${(4 x - 11)}^{3}$ .

$\frac{{(4 x - 11)}^{2} (4 x - 11) - 3 x ({(4 x - 11)}^{2} \frac{d}{d x} [4 x - 11])}{{(4 x - 11)}^{6}}$

Step 4.2.2

Factor ${(4 x - 11)}^{2}$ out of $- 3 x ({(4 x - 11)}^{2} \frac{d}{d x} [4 x - 11])$ .

$\frac{{(4 x - 11)}^{2} (4 x - 11) + {(4 x - 11)}^{2} (- 3 x (\frac{d}{d x} [4 x - 11]))}{{(4 x - 11)}^{6}}$

Step 4.2.3

Factor ${(4 x - 11)}^{2}$ out of ${(4 x - 11)}^{2} (4 x - 11) + {(4 x - 11)}^{2} (- 3 x (\frac{d}{d x} [4 x - 11]))$ .

$\frac{{(4 x - 11)}^{2} (4 x - 11 - 3 x (\frac{d}{d x} [4 x - 11]))}{{(4 x - 11)}^{6}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor ${(4 x - 11)}^{2}$ out of ${(4 x - 11)}^{6}$ .

$\frac{{(4 x - 11)}^{2} (4 x - 11 - 3 x (\frac{d}{d x} [4 x - 11]))}{{(4 x - 11)}^{2} {(4 x - 11)}^{4}}$

Step 5.2

Cancel the common factor.

$\frac{{(4 x - 11)}^{2} (4 x - 11 - 3 x (\frac{d}{d x} [4 x - 11]))}{{(4 x - 11)}^{2} {(4 x - 11)}^{4}}$

Step 5.3

Rewrite the expression.

$\frac{4 x - 11 - 3 x (\frac{d}{d x} [4 x - 11])}{{(4 x - 11)}^{4}}$

Step 6

By the Sum Rule, the derivative of $4 x - 11$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 11]$ .

$\frac{4 x - 11 - 3 x (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 11])}{{(4 x - 11)}^{4}}$

Step 7

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{4 x - 11 - 3 x (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 11])}{{(4 x - 11)}^{4}}$

Step 8

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

$\frac{4 x - 11 - 3 x (4 \cdot 1 + \frac{d}{d x} [- 11])}{{(4 x - 11)}^{4}}$

Step 9

Multiply $4$ by $1$ .

$\frac{4 x - 11 - 3 x (4 + \frac{d}{d x} [- 11])}{{(4 x - 11)}^{4}}$

Step 10

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11$ with respect to $x$ is $0$ .

$\frac{4 x - 11 - 3 x (4 + 0)}{{(4 x - 11)}^{4}}$

Step 11

Simplify by adding terms.

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Step 11.1

Add $4$ and $0$ .

$\frac{4 x - 11 - 3 x \cdot 4}{{(4 x - 11)}^{4}}$

Step 11.2

Multiply $4$ by $- 3$ .

$\frac{4 x - 11 - 12 x}{{(4 x - 11)}^{4}}$

Step 11.3

Subtract $12 x$ from $4 x$ .

$\frac{- 8 x - 11}{{(4 x - 11)}^{4}}$

Step 12

Simplify.

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Step 12.1

Factor $- 1$ out of $- 8 x$ .

$\frac{- (8 x) - 11}{{(4 x - 11)}^{4}}$

Step 12.2

Rewrite $- 11$ as $- 1 (11)$ .

$\frac{- (8 x) - 1 (11)}{{(4 x - 11)}^{4}}$

Step 12.3

Factor $- 1$ out of $- (8 x) - 1 (11)$ .

$\frac{- (8 x + 11)}{{(4 x - 11)}^{4}}$

Step 12.4

Rewrite $- (8 x + 11)$ as $- 1 (8 x + 11)$ .

$\frac{- 1 (8 x + 11)}{{(4 x - 11)}^{4}}$

Step 12.5

Move the negative in front of the fraction.

$- \frac{8 x + 11}{{(4 x - 11)}^{4}}$