Calculus Examples

$p (x) = - \frac{1}{2} \sqrt{x - 3} + 1$

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

By the Sum Rule, the derivative of $- \frac{1}{2} \cdot \sqrt{x - 3} + 1$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{2} \cdot \sqrt{x - 3}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [- \frac{1}{2} \cdot \sqrt{x - 3}] + \frac{d}{d x} [1]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- \frac{1}{2} \cdot \sqrt{x - 3}]$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} \cdot {(x - 3)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [{(x - 3)}^{\frac{1}{2}}]$ .

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

Step 1.1.2.3

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 $g (x) = x - 3$ .

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Step 1.1.2.3.1

To apply the Chain Rule, set $u$ as $x - 3$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

Replace all occurrences of $u$ with $x - 3$ .

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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

Step 1.1.2.9

Combine the numerators over the common denominator.

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

Step 1.1.2.10

Simplify the numerator.

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Step 1.1.2.10.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.10.2

Subtract $2$ from $1$ .

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

Step 1.1.2.11

Move the negative in front of the fraction.

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

Step 1.1.2.12

Add $1$ and $0$ .

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

Step 1.1.2.13

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

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

Step 1.1.2.14

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

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

Step 1.1.2.15

Move ${(x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.2.16

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

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

Step 1.1.2.17

Multiply $2$ by $2$ .

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

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Add $- \frac{1}{4 {(x - 3)}^{\frac{1}{2}}}$ and $0$ .

$f' (x) = - \frac{1}{4 {(x - 3)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $p (x)$ with respect to $x$ is $- \frac{1}{4 {(x - 3)}^{\frac{1}{2}}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{1}{4 {(x - 3)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

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

Step 3.1.2

Anything raised to $1$ is the base itself.

$- \frac{1}{4 \sqrt{x - 3}}$

Step 3.2

Set the denominator in $\frac{1}{4 \sqrt{x - 3}}$ equal to $0$ to find where the expression is undefined.

$4 \sqrt{x - 3} = 0$

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.

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

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

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

Apply the product rule to $4 {(x - 3)}^{\frac{1}{2}}$ .

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

Step 3.3.2.2.1.2

Raise $4$ to the power of $2$ .

$16 {({(x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.3

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

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Step 3.3.2.2.1.3.1

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

$16 {(x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$16 {(x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$16 {(x - 3)}^{1} = 0^{2}$

Step 3.3.2.2.1.4

Simplify.

$16 (x - 3) = 0^{2}$

Step 3.3.2.2.1.5

Apply the distributive property.

$16 x + 16 \cdot - 3 = 0^{2}$

Step 3.3.2.2.1.6

Multiply $16$ by $- 3$ .

$16 x - 48 = 0^{2}$

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 $0$ .

$16 x - 48 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Add $48$ to both sides of the equation.

$16 x = 48$

Step 3.3.3.2

Divide each term in $16 x = 48$ by $16$ and simplify.

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Step 3.3.3.2.1

Divide each term in $16 x = 48$ by $16$ .

$\frac{16 x}{16} = \frac{48}{16}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $16$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{48}{16}$

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{48}{16}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $48$ by $16$ .

$x = 3$

Step 3.4

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

$x - 3 < 0$

Step 3.5

Add $3$ to both sides of the inequality.

$x < 3$

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 $0$ .

$x \leq 3$

$(- \infty, 3]$

$x \leq 3$

$(- \infty, 3]$

Step 4

Evaluate $- \frac{1}{2} \cdot \sqrt{x - 3} + 1$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 3$ .

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Step 4.1.1

Substitute $3$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Subtract $3$ from $3$ .

$- \frac{1}{2} \cdot \sqrt{0} + 1$

Step 4.1.2.1.2

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

$- \frac{1}{2} \cdot \sqrt{0^{2}} + 1$

Step 4.1.2.1.3

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

$- \frac{1}{2} \cdot 0 + 1$

Step 4.1.2.1.4

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 4.1.2.1.4.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.1.4.2

Multiply $0$ by $\frac{1}{2}$ .

$0 + 1$

Step 4.1.2.2

Add $0$ and $1$ .

$1$

Step 4.2

List all of the points.

$(3, 1)$

Step 5