Calculus Examples

$f (x) = \frac{2 x + 3}{\sqrt{4 x - 5}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 5}$ as ${(4 x - 5)}^{\frac{1}{2}}$ .

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

Step 1.1.2

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 $f (x) = 2 x + 3$ and $g (x) = {(4 x - 5)}^{\frac{1}{2}}$ .

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

Step 1.1.3

Multiply the exponents in ${({(4 x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 1.1.3.1

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

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

Step 1.1.3.2

Cancel the common factor of $2$ .

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Step 1.1.3.2.1

Cancel the common factor.

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

Step 1.1.3.2.2

Rewrite the expression.

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

Step 1.1.4

Simplify.

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

Step 1.1.5

Differentiate.

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Step 1.1.5.1

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

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

Step 1.1.5.2

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

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

Step 1.1.5.3

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

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

Step 1.1.5.4

Multiply $2$ by $1$ .

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

Step 1.1.5.5

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

$\frac{{(4 x - 5)}^{\frac{1}{2}} (2 + 0) - (2 x + 3) \frac{d}{d x} [{(4 x - 5)}^{\frac{1}{2}}]}{4 x - 5}$

Step 1.1.5.6

Simplify the expression.

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Step 1.1.5.6.1

Add $2$ and $0$ .

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

Step 1.1.5.6.2

Move $2$ to the left of ${(4 x - 5)}^{\frac{1}{2}}$ .

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

Step 1.1.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 x - 5$ .

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Step 1.1.6.1

To apply the Chain Rule, set $u_{1}$ as $4 x - 5$ .

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

Step 1.1.6.2

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

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

Step 1.1.6.3

Replace all occurrences of $u_{1}$ with $4 x - 5$ .

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

Step 1.1.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.1.8

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 1.1.9

Combine the numerators over the common denominator.

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

Step 1.1.10

Simplify the numerator.

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Step 1.1.10.1

Multiply $- 1$ by $2$ .

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

Step 1.1.10.2

Subtract $2$ from $1$ .

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

Step 1.1.11

Combine fractions.

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Step 1.1.11.1

Move the negative in front of the fraction.

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

Step 1.1.11.2

Combine $\frac{1}{2}$ and ${(4 x - 5)}^{- \frac{1}{2}}$ .

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

Step 1.1.11.3

Move ${(4 x - 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.12

By the Sum Rule, the derivative of $4 x - 5$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5]$ .

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

Step 1.1.13

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

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

Step 1.1.14

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

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

Step 1.1.15

Multiply $4$ by $1$ .

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

Step 1.1.16

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

$\frac{2 {(4 x - 5)}^{\frac{1}{2}} - (2 x + 3) (\frac{1}{2 {(4 x - 5)}^{\frac{1}{2}}} (4 + 0))}{4 x - 5}$

Step 1.1.17

Simplify terms.

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Step 1.1.17.1

Add $4$ and $0$ .

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

Step 1.1.17.2

Combine $\frac{1}{2 {(4 x - 5)}^{\frac{1}{2}}}$ and $4$ .

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

Step 1.1.17.3

Factor $2$ out of $4$ .

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

Step 1.1.18

Cancel the common factors.

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Step 1.1.18.1

Factor $2$ out of $2 {(4 x - 5)}^{\frac{1}{2}}$ .

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

Step 1.1.18.2

Cancel the common factor.

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

Step 1.1.18.3

Rewrite the expression.

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

Step 1.1.19

Simplify.

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Step 1.1.19.1

Apply the distributive property.

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

Step 1.1.19.2

Simplify the numerator.

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Step 1.1.19.2.1

Add parentheses.

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

Step 1.1.19.2.2

Let $u_{2} = {(4 x - 5)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(4 x - 5)}^{\frac{1}{2}}$ .

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Step 1.1.19.2.2.1

Factor $2$ out of $2 u_{2} \cdot u_{2} + 2 (- 2 x - 3)$ .

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Step 1.1.19.2.2.1.1

Factor $2$ out of $2 u_{2} \cdot u_{2}$ .

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

Step 1.1.19.2.2.1.2

Factor $2$ out of $2 (u_{2} \cdot u_{2}) + 2 (- 2 x - 3)$ .

$\frac{\frac{2 (u_{2} \cdot u_{2} - 2 x - 3)}{u_{2}}}{4 x - 5}$

Step 1.1.19.2.2.2

Multiply $u_{2}$ by $u_{2}$ .

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

Step 1.1.19.2.3

Replace all occurrences of $u_{2}$ with ${(4 x - 5)}^{\frac{1}{2}}$ .

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

Step 1.1.19.2.4

Simplify.

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Step 1.1.19.2.4.1

Simplify each term.

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Step 1.1.19.2.4.1.1

Multiply the exponents in ${({(4 x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 1.1.19.2.4.1.1.1

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

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

Step 1.1.19.2.4.1.1.2

Cancel the common factor of $2$ .

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Step 1.1.19.2.4.1.1.2.1

Cancel the common factor.

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

Step 1.1.19.2.4.1.1.2.2

Rewrite the expression.

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

Step 1.1.19.2.4.1.2

Simplify.

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

Step 1.1.19.2.4.2

Subtract $2 x$ from $4 x$ .

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

Step 1.1.19.2.4.3

Subtract $3$ from $- 5$ .

$\frac{\frac{2 (2 x - 8)}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$

Step 1.1.19.2.4.4

Apply the distributive property.

$\frac{\frac{2 (2 x) + 2 \cdot - 8}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$

Step 1.1.19.2.4.5

Multiply $2$ by $2$ .

$\frac{\frac{4 x + 2 \cdot - 8}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$

Step 1.1.19.2.4.6

Multiply $2$ by $- 8$ .

$\frac{\frac{4 x - 16}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$

Step 1.1.19.2.5

Factor $4$ out of $4 x - 16$ .

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Step 1.1.19.2.5.1

Factor $4$ out of $4 x$ .

$\frac{\frac{4 (x) - 16}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$

Step 1.1.19.2.5.2

Factor $4$ out of $- 16$ .

$\frac{\frac{4 x + 4 \cdot - 4}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$

Step 1.1.19.2.5.3

Factor $4$ out of $4 x + 4 \cdot - 4$ .

$\frac{\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$

Step 1.1.19.3

Combine terms.

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Step 1.1.19.3.1

Rewrite $\frac{\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{1}{2}}}}{4 x - 5}$ as a product.

$\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{1}{2}}} \cdot \frac{1}{4 x - 5}$

Step 1.1.19.3.2

Multiply $\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{1}{2}}}$ by $\frac{1}{4 x - 5}$ .

$\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{1}{2}} (4 x - 5)}$

Step 1.1.19.3.3

Multiply ${(4 x - 5)}^{\frac{1}{2}}$ by $4 x - 5$ by adding the exponents.

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Step 1.1.19.3.3.1

Multiply ${(4 x - 5)}^{\frac{1}{2}}$ by $4 x - 5$ .

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Step 1.1.19.3.3.1.1

Raise $4 x - 5$ to the power of $1$ .

$\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{1}{2}} {(4 x - 5)}^{1}}$

Step 1.1.19.3.3.1.2

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

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

Step 1.1.19.3.3.2

Write $1$ as a fraction with a common denominator.

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

Step 1.1.19.3.3.3

Combine the numerators over the common denominator.

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

Step 1.1.19.3.3.4

Add $1$ and $2$ .

$f' (x) = \frac{4 (x - 4)}{{(4 x - 5)}^{\frac{3}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{3}{2}}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{3}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{4 (x - 4)}{{(4 x - 5)}^{\frac{3}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 (x - 4) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $4 (x - 4) = 0$ by $4$ and simplify.

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Step 2.3.1.1

Divide each term in $4 (x - 4) = 0$ by $4$ .

$\frac{4 (x - 4)}{4} = \frac{0}{4}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $4$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{4 (x - 4)}{4} = \frac{0}{4}$

Step 2.3.1.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{0}{4}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $4$ .

$x - 4 = 0$

Step 2.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{4 (x - 4)}{\sqrt{{(4 x - 5)}^{3}}}$

Step 3.2

Set the denominator in $\frac{4 (x - 4)}{\sqrt{{(4 x - 5)}^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{{(4 x - 5)}^{3}} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{{(4 x - 5)}^{3}}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(4 x - 5)}^{3}}$ as ${(4 x - 5)}^{\frac{3}{2}}$ .

${({(4 x - 5)}^{\frac{3}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Multiply the exponents in ${({(4 x - 5)}^{\frac{3}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

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

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

Step 3.3.2.2.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.2.1

Cancel the common factor.

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

Step 3.3.2.2.1.2.2

Rewrite the expression.

${(4 x - 5)}^{3} = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

${(4 x - 5)}^{3} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Set the $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Step 3.3.3.2

Solve for $x$ .

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Step 3.3.3.2.1

Add $5$ to both sides of the equation.

$4 x = 5$

Step 3.3.3.2.2

Divide each term in $4 x = 5$ by $4$ and simplify.

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Step 3.3.3.2.2.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 3.3.3.2.2.2

Simplify the left side.

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Step 3.3.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 3.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 3.4

Set the radicand in $\sqrt{{(4 x - 5)}^{3}}$ less than $0$ to find where the expression is undefined.

${(4 x - 5)}^{3} < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{{(4 x - 5)}^{3}} < \sqrt[3]{0}$

Step 3.5.2

Simplify the equation.

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Step 3.5.2.1

Simplify the left side.

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Step 3.5.2.1.1

Pull terms out from under the radical.

$4 x - 5 < \sqrt[3]{0}$

Step 3.5.2.2

Simplify the right side.

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Step 3.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 3.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$4 x - 5 < \sqrt[3]{0^{3}}$

Step 3.5.2.2.1.2

Pull terms out from under the radical.

$4 x - 5 < 0$

Step 3.5.3

Add $5$ to both sides of the inequality.

$4 x < 5$

Step 3.5.4

Divide each term in $4 x < 5$ by $4$ and simplify.

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Step 3.5.4.1

Divide each term in $4 x < 5$ by $4$ .

$\frac{4 x}{4} < \frac{5}{4}$

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $4$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{5}{4}$

Step 3.5.4.2.1.2

Divide $x$ by $1$ .

$x < \frac{5}{4}$

Step 3.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq \frac{5}{4}$

$(- \infty, \frac{5}{4}]$

$x \leq \frac{5}{4}$

$(- \infty, \frac{5}{4}]$

Step 4

Evaluate $\frac{2 x + 3}{\sqrt{4 x - 5}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 4$ .

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Step 4.1.1

Substitute $4$ for $x$ .

$\frac{2 (4) + 3}{\sqrt{4 (4) - 5}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Multiply $2$ by $4$ .

$\frac{8 + 3}{\sqrt{4 (4) - 5}}$

Step 4.1.2.1.2

Add $8$ and $3$ .

$\frac{11}{\sqrt{4 (4) - 5}}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Multiply $4$ by $4$ .

$\frac{11}{\sqrt{16 - 5}}$

Step 4.1.2.2.2

Subtract $5$ from $16$ .

$\frac{11}{\sqrt{11}}$

Step 4.1.2.3

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{11}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 4.1.2.4

Combine and simplify the denominator.

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Step 4.1.2.4.1

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 4.1.2.4.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{11 \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}}$

Step 4.1.2.4.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{11 \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}}$

Step 4.1.2.4.4

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

$\frac{11 \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Step 4.1.2.4.5

Add $1$ and $1$ .

$\frac{11 \sqrt{11}}{{\sqrt{11}}^{2}}$

Step 4.1.2.4.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 4.1.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{11 \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

Step 4.1.2.4.6.2

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

$\frac{11 \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

Step 4.1.2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{11 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 4.1.2.4.6.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.6.4.1

Cancel the common factor.

$\frac{11 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 4.1.2.4.6.4.2

Rewrite the expression.

$\frac{11 \sqrt{11}}{11^{1}}$

Step 4.1.2.4.6.5

Evaluate the exponent.

$\frac{11 \sqrt{11}}{11}$

Step 4.1.2.5

Cancel the common factor of $11$ .

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Step 4.1.2.5.1

Cancel the common factor.

$\frac{11 \sqrt{11}}{11}$

Step 4.1.2.5.2

Divide $\sqrt{11}$ by $1$ .

$\sqrt{11}$

Step 4.2

Evaluate at $x = \frac{5}{4}$ .

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Step 4.2.1

Substitute $\frac{5}{4}$ for $x$ .

$\frac{2 (\frac{5}{4}) + 3}{\sqrt{4 (\frac{5}{4}) - 5}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 (\frac{5}{4}) + 3}{\sqrt{4 (\frac{5}{4}) - 5}}$

Step 4.2.2.1.2

Rewrite the expression.

$\frac{2 (\frac{5}{4}) + 3}{\sqrt{5 - 5}}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Subtract $5$ from $5$ .

$\frac{2 (\frac{5}{4}) + 3}{\sqrt{0}}$

Step 4.2.2.2.2

Rewrite $0$ as $0^{2}$ .

$\frac{2 (\frac{5}{4}) + 3}{\sqrt{0^{2}}}$

Step 4.2.2.2.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 (\frac{5}{4}) + 3}{0}$

Step 4.2.2.2.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{2 (\frac{5}{4}) + 3}{0}$

Step 4.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(4, \sqrt{11})$

Step 5