Calculus Examples

$f (x) = x \sqrt{x - a}$

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

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

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x - a)}^{\frac{1}{2}}$ .

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

Step 1.1.3

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 $g (x) = x - a$ .

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

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

Step 1.1.8.2

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

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

Step 1.1.8.3

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

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

Step 1.1.8.4

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

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

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

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

Step 1.1.12

Simplify the expression.

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Step 1.1.12.1

Add $1$ and $0$ .

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

Step 1.1.12.2

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

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

Step 1.1.13

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

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

Step 1.1.14

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

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

Step 1.1.15

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

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

Step 1.1.16

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

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

Step 1.1.17

Combine the numerators over the common denominator.

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

Step 1.1.18

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

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Step 1.1.18.1

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

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

Step 1.1.18.2

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

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

Step 1.1.18.3

Combine the numerators over the common denominator.

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

Step 1.1.18.4

Add $1$ and $1$ .

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

Step 1.1.18.5

Divide $2$ by $2$ .

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

Step 1.1.19

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

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

Step 1.1.20

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

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

Step 1.1.21

Simplify.

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Step 1.1.21.1

Apply the distributive property.

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

Step 1.1.21.2

Simplify the numerator.

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Step 1.1.21.2.1

Multiply $- 1$ by $2$ .

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

Step 1.1.21.2.2

Add $x$ and $2 x$ .

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

Step 1.1.21.3

Reorder terms.

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

Step 1.1.21.4

Factor $- 1$ out of $- 2 a$ .

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

Step 1.1.21.5

Factor $- 1$ out of $3 x$ .

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

Step 1.1.21.6

Factor $- 1$ out of $- (2 a) - (- 3 x)$ .

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

Step 1.1.21.7

Rewrite $- (2 a - 3 x)$ as $- 1 (2 a - 3 x)$ .

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

Step 1.1.21.8

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$2 a - 3 x = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $2 a$ from both sides of the equation.

$- 3 x = - 2 a$

Step 2.3.2

Divide each term in $- 3 x = - 2 a$ by $- 3$ and simplify.

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Step 2.3.2.1

Divide each term in $- 3 x = - 2 a$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 2 a}{- 3}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 2 a}{- 3}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 a}{- 3}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{2 a}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

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

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Step 4.1

Evaluate at $x = \frac{2 a}{3}$ .

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Step 4.1.1

Substitute $\frac{2 a}{3}$ for $x$ .

$(\frac{2 a}{3}) \sqrt{(\frac{2 a}{3}) - a}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\frac{2 a}{3} \sqrt{\frac{2 a}{3} - a \cdot \frac{3}{3}}$

Step 4.1.2.2

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

$\frac{2 a}{3} \sqrt{\frac{2 a}{3} + \frac{- a \cdot 3}{3}}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{2 a}{3} \sqrt{\frac{2 a - a \cdot 3}{3}}$

Step 4.1.2.4

Rewrite $\frac{2 a - a \cdot 3}{3}$ in a factored form.

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Step 4.1.2.4.1

Factor $a$ out of $2 a - a \cdot 3$ .

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Step 4.1.2.4.1.1

Factor $a$ out of $2 a$ .

$\frac{2 a}{3} \sqrt{\frac{a \cdot 2 - a \cdot 3}{3}}$

Step 4.1.2.4.1.2

Factor $a$ out of $- a \cdot 3$ .

$\frac{2 a}{3} \sqrt{\frac{a \cdot 2 + a (- 1 \cdot 3)}{3}}$

Step 4.1.2.4.1.3

Factor $a$ out of $a \cdot 2 + a (- 1 \cdot 3)$ .

$\frac{2 a}{3} \sqrt{\frac{a (2 - 1 \cdot 3)}{3}}$

Step 4.1.2.4.2

Multiply $- 1$ by $3$ .

$\frac{2 a}{3} \sqrt{\frac{a (2 - 3)}{3}}$

Step 4.1.2.4.3

Subtract $3$ from $2$ .

$\frac{2 a}{3} \sqrt{\frac{a \cdot - 1}{3}}$

Step 4.1.2.5

Simplify the expression.

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Step 4.1.2.5.1

Move $- 1$ to the left of $a$ .

$\frac{2 a}{3} \sqrt{\frac{- 1 \cdot a}{3}}$

Step 4.1.2.5.2

Move the negative in front of the fraction.

$\frac{2 a}{3} \sqrt{- \frac{a}{3}}$

Step 4.1.2.6

Combine $\frac{2 a}{3}$ and $\sqrt{- \frac{a}{3}}$ .

$\frac{2 a \sqrt{- \frac{a}{3}}}{3}$

Step 4.2

List all of the points.

$(\frac{2 a}{3}, \frac{2 a \sqrt{- \frac{a}{3}}}{3})$

Step 5