Calculus Examples

$y = \frac{x}{\sqrt{2 x - 1}}$

Step 1

Simplify $\frac{x}{\sqrt{2 x - 1}}$ .

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Step 1.1

Multiply $\frac{x}{\sqrt{2 x - 1}}$ by $\frac{\sqrt{2 x - 1}}{\sqrt{2 x - 1}}$ .

$y = \frac{x}{\sqrt{2 x - 1}} \cdot \frac{\sqrt{2 x - 1}}{\sqrt{2 x - 1}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{x}{\sqrt{2 x - 1}}$ by $\frac{\sqrt{2 x - 1}}{\sqrt{2 x - 1}}$ .

$y = \frac{x \sqrt{2 x - 1}}{\sqrt{2 x - 1} \sqrt{2 x - 1}}$

Step 1.2.2

Raise $\sqrt{2 x - 1}$ to the power of $1$ .

$y = \frac{x \sqrt{2 x - 1}}{{\sqrt{2 x - 1}}^{1} \sqrt{2 x - 1}}$

Step 1.2.3

Raise $\sqrt{2 x - 1}$ to the power of $1$ .

$y = \frac{x \sqrt{2 x - 1}}{{\sqrt{2 x - 1}}^{1} {\sqrt{2 x - 1}}^{1}}$

Step 1.2.4

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

$y = \frac{x \sqrt{2 x - 1}}{{\sqrt{2 x - 1}}^{1 + 1}}$

Step 1.2.5

Add $1$ and $1$ .

$y = \frac{x \sqrt{2 x - 1}}{{\sqrt{2 x - 1}}^{2}}$

Step 1.2.6

Rewrite ${\sqrt{2 x - 1}}^{2}$ as $2 x - 1$ .

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Step 1.2.6.1

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

$y = \frac{x \sqrt{2 x - 1}}{{({(2 x - 1)}^{\frac{1}{2}})}^{2}}$

Step 1.2.6.2

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

$y = \frac{x \sqrt{2 x - 1}}{{(2 x - 1)}^{\frac{1}{2} \cdot 2}}$

Step 1.2.6.3

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

$y = \frac{x \sqrt{2 x - 1}}{{(2 x - 1)}^{\frac{2}{2}}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$y = \frac{x \sqrt{2 x - 1}}{{(2 x - 1)}^{\frac{2}{2}}}$

Step 1.2.6.4.2

Rewrite the expression.

$y = \frac{x \sqrt{2 x - 1}}{{(2 x - 1)}^{1}}$

Step 1.2.6.5

Simplify.

$y = \frac{x \sqrt{2 x - 1}}{2 x - 1}$

Step 2

Set $y$ as a function of $x$ .

$f (x) = \frac{x \sqrt{2 x - 1}}{2 x - 1}$

Step 3

Find the derivative.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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Step 3.3.1.1

Raise $2 x - 1$ to the power of $1$ .

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

Step 3.3.1.2

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

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

Step 3.3.2

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

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

Step 3.3.3

Combine the numerators over the common denominator.

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

Step 3.3.4

Subtract $1$ from $2$ .

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

Step 3.4

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 $g (x) = {(2 x - 1)}^{\frac{1}{2}}$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Cancel the common factor of $2$ .

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Step 3.5.2.1

Cancel the common factor.

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

Step 3.5.2.2

Rewrite the expression.

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

Step 3.6

Simplify.

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

Step 3.7

Differentiate using the Power Rule.

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.8

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 $g (x) = 2 x - 1$ .

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Step 3.8.1

To apply the Chain Rule, set $u$ as $2 x - 1$ .

$\frac{{(2 x - 1)}^{\frac{1}{2}} - x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [2 x - 1])}{2 x - 1}$

Step 3.8.2

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

$\frac{{(2 x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [2 x - 1])}{2 x - 1}$

Step 3.8.3

Replace all occurrences of $u$ with $2 x - 1$ .

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Combine the numerators over the common denominator.

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

Step 3.12

Simplify the numerator.

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Step 3.12.1

Multiply $- 1$ by $2$ .

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

Step 3.12.2

Subtract $2$ from $1$ .

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

Step 3.13

Combine fractions.

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Step 3.13.1

Move the negative in front of the fraction.

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

Step 3.13.2

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

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

Step 3.13.3

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

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

Step 3.13.4

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

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

Step 3.14

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

Step 3.15

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

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

Step 3.16

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

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

Step 3.17

Multiply $2$ by $1$ .

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

Step 3.18

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 3.19

Simplify terms.

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Step 3.19.1

Add $2$ and $0$ .

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

Step 3.19.2

Multiply $2$ by $- 1$ .

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

Step 3.19.3

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

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

Step 3.19.4

Factor $2$ out of $- 2 x$ .

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

Step 3.20

Cancel the common factors.

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Step 3.20.1

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

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

Step 3.20.2

Cancel the common factor.

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

Step 3.20.3

Rewrite the expression.

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

Step 3.21

Move the negative in front of the fraction.

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

Step 3.22

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

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

Step 3.23

Combine the numerators over the common denominator.

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

Step 3.24

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

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Step 3.24.1

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

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

Step 3.24.2

Combine the numerators over the common denominator.

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

Step 3.24.3

Add $1$ and $1$ .

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

Step 3.24.4

Divide $2$ by $2$ .

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

Step 3.25

Simplify ${(2 x - 1)}^{1}$ .

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

Step 3.26

Subtract $x$ from $2 x$ .

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

Step 3.27

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

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

Step 3.28

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

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

Step 3.29

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

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Step 3.29.1

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

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Step 3.29.1.1

Raise $2 x - 1$ to the power of $1$ .

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

Step 3.29.1.2

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

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

Step 3.29.2

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

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

Step 3.29.3

Combine the numerators over the common denominator.

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

Step 3.29.4

Add $1$ and $2$ .

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

Step 4

Set the derivative equal to $0$ then solve the equation $\frac{x - 1}{{(2 x - 1)}^{\frac{3}{2}}} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$x - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5

Solve the original function $f (x) = \frac{x \sqrt{2 x - 1}}{2 x - 1}$ at $x = 1$ .

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{(1) \sqrt{2 (1) - 1}}{2 (1) - 1}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $\sqrt{2 (1) - 1}$ by $1$ .

$f (1) = \frac{\sqrt{2 (1) - 1}}{2 (1) - 1}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Multiply $2$ by $1$ .

$f (1) = \frac{\sqrt{2 (1) - 1}}{2 - 1}$

Step 5.2.2.2

Subtract $1$ from $2$ .

$f (1) = \frac{\sqrt{2 (1) - 1}}{1}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Multiply $2$ by $1$ .

$f (1) = \frac{\sqrt{2 - 1}}{1}$

Step 5.2.3.2

Subtract $1$ from $2$ .

$f (1) = \frac{\sqrt{1}}{1}$

Step 5.2.3.3

Any root of $1$ is $1$ .

$f (1) = \frac{1}{1}$

Step 5.2.4

Divide $1$ by $1$ .

$f (1) = 1$

Step 5.2.5

The final answer is $1$ .

$1$

Step 6

The horizontal tangent line on function $f (x) = \frac{x \sqrt{2 x - 1}}{2 x - 1}$ is $y = 1$ .

$y = 1$

Step 7