Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1.1.1

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

$f' (x) = \frac{d}{d x} (\frac{2 x}{{(x - 1)}^{\frac{1}{2}}})$

Step 1.1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x}{{(x - 1)}^{\frac{1}{2}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x}{{(x - 1)}^{\frac{1}{2}}}]$ .

$f' (x) = 2 \frac{d}{d x} (\frac{x}{{(x - 1)}^{\frac{1}{2}}})$

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(x - 1)}^{\frac{1}{2}}}{{({(x - 1)}^{\frac{1}{2}})}^{2}})$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(x - 1)}^{\frac{1}{2}}}{{(x - 1)}^{\frac{1}{2} \cdot 2}})$

Step 1.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.3.2.1

Cancel the common factor.

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(x - 1)}^{\frac{1}{2}}}{{(x - 1)}^{\frac{1}{2} \cdot 2}})$

Step 1.1.3.2.2

Rewrite the expression.

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(x - 1)}^{\frac{1}{2}}}{x - 1})$

Step 1.1.4

Simplify.

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(x - 1)}^{\frac{1}{2}}}{x - 1})$

Step 1.1.5

Differentiate using the Power Rule.

Tap for more steps...

Step 1.1.5.1

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} \cdot 1 - x \frac{d}{d x} {(x - 1)}^{\frac{1}{2}}}{x - 1})$

Step 1.1.5.2

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x \frac{d}{d x} {(x - 1)}^{\frac{1}{2}}}{x - 1})$

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 $g (x) = x - 1$ .

Tap for more steps...

Step 1.1.6.1

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.6.2

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.6.3

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(x - 1)}^{\frac{1}{2} - 1} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.7

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(x - 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.8

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(x - 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.9

Combine the numerators over the common denominator.

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(x - 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.10

Simplify the numerator.

Tap for more steps...

Step 1.1.10.1

Multiply $- 1$ by $2$ .

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(x - 1)}^{\frac{1 - 2}{2}} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.10.2

Subtract $2$ from $1$ .

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(x - 1)}^{\frac{- 1}{2}} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.11

Combine fractions.

Tap for more steps...

Step 1.1.11.1

Move the negative in front of the fraction.

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(x - 1)}^{- \frac{1}{2}} \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.11.2

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{{(x - 1)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.11.3

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - x (\frac{1}{2 {(x - 1)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x - 1))}{x - 1})$

Step 1.1.11.4

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - \frac{x}{2 {(x - 1)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x - 1)}{x - 1})$

Step 1.1.12

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - \frac{x}{2 {(x - 1)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (- 1))}{x - 1})$

Step 1.1.13

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - \frac{x}{2 {(x - 1)}^{\frac{1}{2}}} \cdot (1 + \frac{d}{d x} (- 1))}{x - 1})$

Step 1.1.14

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

$f' (x) = 2 (\frac{{(x - 1)}^{\frac{1}{2}} - \frac{x}{2 {(x - 1)}^{\frac{1}{2}}} \cdot (1 + 0)}{x - 1})$

Step 1.1.15

Simplify the expression.

Tap for more steps...

Step 1.1.15.1

Add $1$ and $0$ .

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

Step 1.1.15.2

Multiply $- 1$ by $1$ .

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

Step 1.1.16

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

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

Step 1.1.17

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

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

Step 1.1.18

Combine the numerators over the common denominator.

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

Step 1.1.19

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

Tap for more steps...

Step 1.1.19.1

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

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

Step 1.1.19.2

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

$f' (x) = 2 (\frac{\frac{{(x - 1)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2 - x}{2 {(x - 1)}^{\frac{1}{2}}}}{x - 1})$

Step 1.1.19.3

Combine the numerators over the common denominator.

$f' (x) = 2 (\frac{\frac{{(x - 1)}^{\frac{1 + 1}{2}} \cdot 2 - x}{2 {(x - 1)}^{\frac{1}{2}}}}{x - 1})$

Step 1.1.19.4

Add $1$ and $1$ .

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

Step 1.1.19.5

Divide $2$ by $2$ .

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

Step 1.1.20

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

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

Step 1.1.21

Move $2$ to the left of $x - 1$ .

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

Step 1.1.22

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

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

Step 1.1.23

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

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

Step 1.1.24

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

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

Step 1.1.25

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

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

Step 1.1.26

Simplify the expression.

Tap for more steps...

Step 1.1.26.1

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

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

Step 1.1.26.2

Combine the numerators over the common denominator.

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

Step 1.1.26.3

Add $2$ and $1$ .

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

Step 1.1.27

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

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

Step 1.1.28

Cancel the common factor.

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

Step 1.1.29

Rewrite the expression.

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

Step 1.1.30

Simplify.

Tap for more steps...

Step 1.1.30.1

Apply the distributive property.

$f' (x) = \frac{2 x + 2 \cdot - 1 - x}{{(x - 1)}^{\frac{3}{2}}}$

Step 1.1.30.2

Simplify the numerator.

Tap for more steps...

Step 1.1.30.2.1

Multiply $2$ by $- 1$ .

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

Step 1.1.30.2.2

Subtract $x$ from $2 x$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x - 2}{{(x - 1)}^{\frac{3}{2}}}$ .

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

Step 2

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

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$x - 2 = 0$

Step 2.3

Add $2$ to both sides of the equation.

$x = 2$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Set the denominator in $\frac{x - 2}{\sqrt{{(x - 1)}^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{{(x - 1)}^{3}} = 0$

Step 3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.1

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

${\sqrt{{(x - 1)}^{3}}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

Tap for more steps...

Step 3.3.2.1

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

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

Step 3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.2.1

Multiply the exponents in ${({(x - 1)}^{\frac{3}{2}})}^{2}$ .

Tap for more steps...

Step 3.3.2.2.1.1

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

${(x - 1)}^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.2.1.2.1

Cancel the common factor.

${(x - 1)}^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.2.2

Rewrite the expression.

${(x - 1)}^{3} = 0^{2}$

Step 3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.2.3.1

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

${(x - 1)}^{3} = 0$

Step 3.3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.3.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.3.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.4

Set the radicand in $\sqrt{{(x - 1)}^{3}}$ less than $0$ to find where the expression is undefined.

${(x - 1)}^{3} < 0$

Step 3.5

Solve for $x$ .

Tap for more steps...

Step 3.5.1

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

$\sqrt[3]{{(x - 1)}^{3}} < \sqrt[3]{0}$

Step 3.5.2

Simplify the equation.

Tap for more steps...

Step 3.5.2.1

Simplify the left side.

Tap for more steps...

Step 3.5.2.1.1

Pull terms out from under the radical.

$x - 1 < \sqrt[3]{0}$

Step 3.5.2.2

Simplify the right side.

Tap for more steps...

Step 3.5.2.2.1

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

Tap for more steps...

Step 3.5.2.2.1.1

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

$x - 1 < \sqrt[3]{0^{3}}$

Step 3.5.2.2.1.2

Pull terms out from under the radical.

$x - 1 < 0$

Step 3.5.3

Add $1$ to both sides of the inequality.

$x < 1$

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 $0$ .

$x \leq 1$

$(- \infty, 1]$

$x \leq 1$

$(- \infty, 1]$

Step 4

Evaluate $\frac{2 x}{\sqrt{x - 1}}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 2$ .

Tap for more steps...

Step 4.1.1

Substitute $2$ for $x$ .

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

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Multiply $2$ by $2$ .

$\frac{4}{\sqrt{2 - 1}}$

Step 4.1.2.2

Simplify the denominator.

Tap for more steps...

Step 4.1.2.2.1

Subtract $1$ from $2$ .

$\frac{4}{\sqrt{1}}$

Step 4.1.2.2.2

Any root of $1$ is $1$ .

$\frac{4}{1}$

Step 4.1.2.3

Divide $4$ by $1$ .

$4$

Step 4.2

Evaluate at $x = 1$ .

Tap for more steps...

Step 4.2.1

Substitute $1$ for $x$ .

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

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Remove parentheses.

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

Step 4.2.2.2

Subtract $1$ from $1$ .

$\frac{2 (1)}{\sqrt{0}}$

Step 4.2.2.3

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

$\frac{2 (1)}{\sqrt{0^{2}}}$

Step 4.2.2.4

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

$\frac{2 (1)}{0}$

Step 4.2.2.5

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

Undefined

Step 4.3

List all of the points.

$(2, 4)$

Step 5