Calculus Examples

${(\frac{3 x - 7}{6 x + 3})}^{4}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \frac{3 x - 7}{6 x + 3}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{3 x - 7}{6 x + 3}$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [\frac{3 x - 7}{6 x + 3}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$4 u^{3} \frac{d}{d x} [\frac{3 x - 7}{6 x + 3}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{3 x - 7}{6 x + 3}$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{d}{d x} [\frac{3 x - 7}{6 x + 3}]$

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 $f (x) = 3 x - 7$ and $g (x) = 6 x + 3$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{(6 x + 3) \frac{d}{d x} [3 x - 7] - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x - 7$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 7]$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{(6 x + 3) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3.2

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

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{(6 x + 3) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3.3

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

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{(6 x + 3) (3 \cdot 1 + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3.4

Multiply $3$ by $1$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{(6 x + 3) (3 + \frac{d}{d x} [- 7]) - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{(6 x + 3) (3 + 0) - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{(6 x + 3) \cdot 3 - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3.6.2

Move $3$ to the left of $6 x + 3$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 \cdot (6 x + 3) - (3 x - 7) \frac{d}{d x} [6 x + 3]}{{(6 x + 3)}^{2}}$

Step 3.7

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

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x + 3) - (3 x - 7) (\frac{d}{d x} [6 x] + \frac{d}{d x} [3])}{{(6 x + 3)}^{2}}$

Step 3.8

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x + 3) - (3 x - 7) (6 \frac{d}{d x} [x] + \frac{d}{d x} [3])}{{(6 x + 3)}^{2}}$

Step 3.9

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

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x + 3) - (3 x - 7) (6 \cdot 1 + \frac{d}{d x} [3])}{{(6 x + 3)}^{2}}$

Step 3.10

Multiply $6$ by $1$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x + 3) - (3 x - 7) (6 + \frac{d}{d x} [3])}{{(6 x + 3)}^{2}}$

Step 3.11

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x + 3) - (3 x - 7) (6 + 0)}{{(6 x + 3)}^{2}}$

Step 3.12

Simplify the expression.

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Step 3.12.1

Add $6$ and $0$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x + 3) - (3 x - 7) \cdot 6}{{(6 x + 3)}^{2}}$

Step 3.12.2

Multiply $6$ by $- 1$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x + 3) - 6 (3 x - 7)}{{(6 x + 3)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x) + 3 \cdot 3 - 6 (3 x - 7)}{{(6 x + 3)}^{2}}$

Step 4.2

Apply the distributive property.

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (6 x) + 3 \cdot 3 - 6 (3 x) - 6 \cdot - 7}{{(6 x + 3)}^{2}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $6$ by $3$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{18 x + 3 \cdot 3 - 6 (3 x) - 6 \cdot - 7}{{(6 x + 3)}^{2}}$

Step 4.3.1.2

Multiply $3$ by $3$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{18 x + 9 - 6 (3 x) - 6 \cdot - 7}{{(6 x + 3)}^{2}}$

Step 4.3.1.3

Multiply $3$ by $- 6$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{18 x + 9 - 18 x - 6 \cdot - 7}{{(6 x + 3)}^{2}}$

Step 4.3.1.4

Multiply $- 6$ by $- 7$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{18 x + 9 - 18 x + 42}{{(6 x + 3)}^{2}}$

Step 4.3.2

Combine the opposite terms in $18 x + 9 - 18 x + 42$ .

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Step 4.3.2.1

Subtract $18 x$ from $18 x$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{0 + 9 + 42}{{(6 x + 3)}^{2}}$

Step 4.3.2.2

Add $0$ and $9$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{9 + 42}{{(6 x + 3)}^{2}}$

Step 4.3.3

Add $9$ and $42$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{51}{{(6 x + 3)}^{2}}$

Step 4.4

Simplify the denominator.

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Step 4.4.1

Factor $3$ out of $6 x + 3$ .

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Step 4.4.1.1

Factor $3$ out of $6 x$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{51}{{(3 (2 x) + 3)}^{2}}$

Step 4.4.1.2

Factor $3$ out of $3$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{51}{{(3 (2 x) + 3 (1))}^{2}}$

Step 4.4.1.3

Factor $3$ out of $3 (2 x) + 3 (1)$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{51}{{(3 (2 x + 1))}^{2}}$

Step 4.4.2

Apply the product rule to $3 (2 x + 1)$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{51}{3^{2} {(2 x + 1)}^{2}}$

Step 4.4.3

Raise $3$ to the power of $2$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{51}{9 {(2 x + 1)}^{2}}$

Step 4.5

Cancel the common factor of $51$ and $9$ .

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Step 4.5.1

Factor $3$ out of $51$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (17)}{9 {(2 x + 1)}^{2}}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $3$ out of $9 {(2 x + 1)}^{2}$ .

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 (17)}{3 (3 {(2 x + 1)}^{2})}$

Step 4.5.2.2

Cancel the common factor.

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{3 \cdot 17}{3 (3 {(2 x + 1)}^{2})}$

Step 4.5.2.3

Rewrite the expression.

$4 {(\frac{3 x - 7}{6 x + 3})}^{3} \frac{17}{3 {(2 x + 1)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $\frac{3 x - 7}{6 x + 3}$ .

$4 \frac{{(3 x - 7)}^{3}}{{(6 x + 3)}^{3}} \frac{17}{3 {(2 x + 1)}^{2}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $4$ and $\frac{{(3 x - 7)}^{3}}{{(6 x + 3)}^{3}}$ .

$\frac{4 {(3 x - 7)}^{3}}{{(6 x + 3)}^{3}} \cdot \frac{17}{3 {(2 x + 1)}^{2}}$

Step 5.2.2

Multiply $\frac{4 {(3 x - 7)}^{3}}{{(6 x + 3)}^{3}}$ by $\frac{17}{3 {(2 x + 1)}^{2}}$ .

$\frac{4 {(3 x - 7)}^{3} \cdot 17}{{(6 x + 3)}^{3} (3 {(2 x + 1)}^{2})}$

Step 5.2.3

Multiply $17$ by $4$ .

$\frac{68 {(3 x - 7)}^{3}}{{(6 x + 3)}^{3} (3 {(2 x + 1)}^{2})}$

Step 5.2.4

Move $3$ to the left of ${(6 x + 3)}^{3}$ .

$\frac{68 {(3 x - 7)}^{3}}{3 {(6 x + 3)}^{3} {(2 x + 1)}^{2}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Factor $3$ out of $6 x + 3$ .

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Step 5.3.1.1

Factor $3$ out of $6 x$ .

$\frac{68 {(3 x - 7)}^{3}}{3 {(3 (2 x) + 3)}^{3} {(2 x + 1)}^{2}}$

Step 5.3.1.2

Factor $3$ out of $3$ .

$\frac{68 {(3 x - 7)}^{3}}{3 {(3 (2 x) + 3 (1))}^{3} {(2 x + 1)}^{2}}$

Step 5.3.1.3

Factor $3$ out of $3 (2 x) + 3 (1)$ .

$\frac{68 {(3 x - 7)}^{3}}{3 {(3 (2 x + 1))}^{3} {(2 x + 1)}^{2}}$

Step 5.3.2

Apply the product rule to $3 (2 x + 1)$ .

$\frac{68 {(3 x - 7)}^{3}}{3 \cdot 3^{3} {(2 x + 1)}^{3} {(2 x + 1)}^{2}}$

Step 5.3.3

Combine exponents.

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Step 5.3.3.1

Multiply $3$ by $3^{3}$ by adding the exponents.

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Step 5.3.3.1.1

Multiply $3$ by $3^{3}$ .

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Step 5.3.3.1.1.1

Raise $3$ to the power of $1$ .

$\frac{68 {(3 x - 7)}^{3}}{3^{1} \cdot 3^{3} {(2 x + 1)}^{3} {(2 x + 1)}^{2}}$

Step 5.3.3.1.1.2

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

$\frac{68 {(3 x - 7)}^{3}}{3^{1 + 3} {(2 x + 1)}^{3} {(2 x + 1)}^{2}}$

Step 5.3.3.1.2

Add $1$ and $3$ .

$\frac{68 {(3 x - 7)}^{3}}{3^{4} {(2 x + 1)}^{3} {(2 x + 1)}^{2}}$

Step 5.3.3.2

Multiply ${(2 x + 1)}^{3}$ by ${(2 x + 1)}^{2}$ by adding the exponents.

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Step 5.3.3.2.1

Move ${(2 x + 1)}^{2}$ .

$\frac{68 {(3 x - 7)}^{3}}{3^{4} ({(2 x + 1)}^{2} {(2 x + 1)}^{3})}$

Step 5.3.3.2.2

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

$\frac{68 {(3 x - 7)}^{3}}{3^{4} {(2 x + 1)}^{2 + 3}}$

Step 5.3.3.2.3

Add $2$ and $3$ .

$\frac{68 {(3 x - 7)}^{3}}{3^{4} {(2 x + 1)}^{5}}$

Step 5.3.4

Raise $3$ to the power of $4$ .

$\frac{68 {(3 x - 7)}^{3}}{81 {(2 x + 1)}^{5}}$