Calculus Examples

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

Step 1

Find the derivative.

Tap for more steps...

Step 1.1

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

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

Step 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 $f (x) = x$ 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 1.3

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

Tap for more steps...

Step 1.3.1

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 1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.2.1

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 1.3.2.2

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 1.4

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 1.5

Differentiate using the Power Rule.

Tap for more steps...

Step 1.5.1

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 1.5.2

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

Tap for more steps...

Step 1.6.1

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 1.6.2

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 1.6.3

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 1.7

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 1.8

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 1.9

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 1.10

Simplify the numerator.

Tap for more steps...

Step 1.10.1

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 1.10.2

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 1.11

Combine fractions.

Tap for more steps...

Step 1.11.1

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 1.11.2

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 1.11.3

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 1.11.4

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 1.12

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 1.13

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 1.14

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

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

Step 1.15

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 1.16

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 1.17

Simplify terms.

Tap for more steps...

Step 1.17.1

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 1.17.2

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 1.17.3

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 1.17.4

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 1.18

Cancel the common factors.

Tap for more steps...

Step 1.18.1

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 1.18.2

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 1.18.3

Rewrite the expression.

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

Step 1.19

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 1.20

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 1.21

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 1.22

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

Tap for more steps...

Step 1.22.1

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 1.22.2

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 1.22.3

Add $1$ and $1$ .

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

Step 1.22.4

Divide $2$ by $2$ .

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

Step 1.23

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

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

Step 1.24

Subtract $x$ from $2 x$ .

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

Step 1.25

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 1.26

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 1.27

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

Tap for more steps...

Step 1.27.1

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

Tap for more steps...

Step 1.27.1.1

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

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

Step 1.27.1.2

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 1.27.2

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

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

Step 1.27.3

Combine the numerators over the common denominator.

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

Step 1.27.4

Add $1$ and $2$ .

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

Step 2

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

Tap for more steps...

Step 2.1

Set the numerator equal to zero.

$x - 1 = 0$

Step 2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify the denominator.

Tap for more steps...

Step 3.2.1.1

Multiply $2$ by $1$ .

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

Step 3.2.1.2

Subtract $1$ from $2$ .

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

Step 3.2.1.3

Any root of $1$ is $1$ .

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

Step 3.2.2

Divide $1$ by $1$ .

$f (1) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 4

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

$y = 1$

Step 5