Calculus Examples

$\sqrt[3]{\frac{2 x + 5}{3 x + 2}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{2 x + 5}{3 x + 2}}$ as ${(\frac{2 x + 5}{3 x + 2})}^{\frac{1}{3}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = \frac{2 x + 5}{3 x + 2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{2 x + 5}{3 x + 2}$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [\frac{2 x + 5}{3 x + 2}]$

Step 2.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [\frac{2 x + 5}{3 x + 2}]$

Step 2.3

Replace all occurrences of $u$ with $\frac{2 x + 5}{3 x + 2}$ .

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

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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 $f (x) = 2 x + 5$ and $g (x) = 3 x + 2$ .

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

Step 9

Differentiate.

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Step 9.1

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

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

Step 9.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 9.3

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

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

Step 9.4

Multiply $2$ by $1$ .

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

Step 9.5

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{3} {(\frac{2 x + 5}{3 x + 2})}^{- \frac{2}{3}} \frac{(3 x + 2) (2 + 0) - (2 x + 5) \frac{d}{d x} [3 x + 2]}{{(3 x + 2)}^{2}}$

Step 9.6

Simplify the expression.

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Step 9.6.1

Add $2$ and $0$ .

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

Step 9.6.2

Move $2$ to the left of $3 x + 2$ .

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

Step 9.7

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

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

Step 9.8

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

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

Step 9.9

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

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

Step 9.10

Multiply $3$ by $1$ .

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

Step 9.11

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{3} {(\frac{2 x + 5}{3 x + 2})}^{- \frac{2}{3}} \frac{2 (3 x + 2) - (2 x + 5) (3 + 0)}{{(3 x + 2)}^{2}}$

Step 9.12

Combine fractions.

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Step 9.12.1

Add $3$ and $0$ .

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

Step 9.12.2

Multiply $3$ by $- 1$ .

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

Step 9.12.3

Multiply $\frac{2 (3 x + 2) - 3 (2 x + 5)}{{(3 x + 2)}^{2}}$ by $\frac{1}{3}$ .

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

Step 9.12.4

Move $3$ to the left of ${(3 x + 2)}^{2}$ .

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

Step 10

Simplify.

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Step 10.1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 10.2

Apply the product rule to $\frac{3 x + 2}{2 x + 5}$ .

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

Step 10.3

Apply the distributive property.

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

Step 10.4

Apply the distributive property.

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

Step 10.5

Combine terms.

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Step 10.5.1

Multiply $3$ by $2$ .

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

Step 10.5.2

Multiply $2$ by $2$ .

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

Step 10.5.3

Multiply $2$ by $- 3$ .

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

Step 10.5.4

Multiply $- 3$ by $5$ .

$\frac{6 x + 4 - 6 x - 15}{3 {(3 x + 2)}^{2}} \cdot \frac{{(3 x + 2)}^{\frac{2}{3}}}{{(2 x + 5)}^{\frac{2}{3}}}$

Step 10.5.5

Subtract $6 x$ from $6 x$ .

$\frac{0 + 4 - 15}{3 {(3 x + 2)}^{2}} \cdot \frac{{(3 x + 2)}^{\frac{2}{3}}}{{(2 x + 5)}^{\frac{2}{3}}}$

Step 10.5.6

Add $0$ and $4$ .

$\frac{4 - 15}{3 {(3 x + 2)}^{2}} \cdot \frac{{(3 x + 2)}^{\frac{2}{3}}}{{(2 x + 5)}^{\frac{2}{3}}}$

Step 10.5.7

Subtract $15$ from $4$ .

$\frac{- 11}{3 {(3 x + 2)}^{2}} \cdot \frac{{(3 x + 2)}^{\frac{2}{3}}}{{(2 x + 5)}^{\frac{2}{3}}}$

Step 10.5.8

Move the negative in front of the fraction.

$- \frac{11}{3 {(3 x + 2)}^{2}} \cdot \frac{{(3 x + 2)}^{\frac{2}{3}}}{{(2 x + 5)}^{\frac{2}{3}}}$

Step 10.5.9

Multiply $\frac{{(3 x + 2)}^{\frac{2}{3}}}{{(2 x + 5)}^{\frac{2}{3}}}$ by $\frac{11}{3 {(3 x + 2)}^{2}}$ .

$- \frac{{(3 x + 2)}^{\frac{2}{3}} \cdot 11}{{(2 x + 5)}^{\frac{2}{3}} (3 {(3 x + 2)}^{2})}$

Step 10.5.10

Move $11$ to the left of ${(3 x + 2)}^{\frac{2}{3}}$ .

$- \frac{11 \cdot {(3 x + 2)}^{\frac{2}{3}}}{{(2 x + 5)}^{\frac{2}{3}} (3 {(3 x + 2)}^{2})}$

Step 10.5.11

Move $3$ to the left of ${(2 x + 5)}^{\frac{2}{3}}$ .

$- \frac{11 {(3 x + 2)}^{\frac{2}{3}}}{3 \cdot {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{2}}$

Step 10.5.12

Move ${(3 x + 2)}^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{2} {(3 x + 2)}^{- \frac{2}{3}}}$

Step 10.5.13

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

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Step 10.5.13.1

Move ${(3 x + 2)}^{- \frac{2}{3}}$ .

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} ({(3 x + 2)}^{- \frac{2}{3}} {(3 x + 2)}^{2})}$

Step 10.5.13.2

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

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{- \frac{2}{3} + 2}}$

Step 10.5.13.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{- \frac{2}{3} + 2 \cdot \frac{3}{3}}}$

Step 10.5.13.4

Combine $2$ and $\frac{3}{3}$ .

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{- \frac{2}{3} + \frac{2 \cdot 3}{3}}}$

Step 10.5.13.5

Combine the numerators over the common denominator.

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{\frac{- 2 + 2 \cdot 3}{3}}}$

Step 10.5.13.6

Simplify the numerator.

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Step 10.5.13.6.1

Multiply $2$ by $3$ .

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{\frac{- 2 + 6}{3}}}$

Step 10.5.13.6.2

Add $- 2$ and $6$ .

$- \frac{11}{3 {(2 x + 5)}^{\frac{2}{3}} {(3 x + 2)}^{\frac{4}{3}}}$