Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $4 + \frac{1}{3} x - \frac{1}{2} x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [\frac{1}{3} x] + \frac{d}{d x} [- \frac{1}{2} x^{2}]$ .

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

Step 1.1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x]$ .

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Step 1.1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Multiply $\frac{1}{3}$ by $1$ .

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

Step 1.1.3

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{2}]$ .

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.1.3.4

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

$0 + \frac{1}{3} + \frac{- 2}{2} x$

Step 1.1.3.5

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

$0 + \frac{1}{3} + \frac{- 2 x}{2}$

Step 1.1.3.6

Cancel the common factor of $- 2$ and $2$ .

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Step 1.1.3.6.1

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

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

Step 1.1.3.6.2

Cancel the common factors.

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Step 1.1.3.6.2.1

Factor $2$ out of $2$ .

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

Step 1.1.3.6.2.2

Cancel the common factor.

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

Step 1.1.3.6.2.3

Rewrite the expression.

$0 + \frac{1}{3} + \frac{- x}{1}$

Step 1.1.3.6.2.4

Divide $- x$ by $1$ .

$0 + \frac{1}{3} - x$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Add $0$ and $\frac{1}{3}$ .

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

Step 1.1.4.2

Reorder terms.

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

Step 1.2

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

$- x + \frac{1}{3}$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$- x + \frac{1}{3} = 0$

Step 2.2

Subtract $\frac{1}{3}$ from both sides of the equation.

$- x = - \frac{1}{3}$

Step 2.3

Divide each term in $- x = - \frac{1}{3}$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- x = - \frac{1}{3}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- \frac{1}{3}}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- \frac{1}{3}}{- 1}$

Step 2.3.2.2

Divide $x$ by $1$ .

$x = \frac{- \frac{1}{3}}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{\frac{1}{3}}{1}$

Step 2.3.3.2

Divide $\frac{1}{3}$ by $1$ .

$x = \frac{1}{3}$

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 $4 + \frac{1}{3} x - \frac{1}{2} x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{1}{3}$ .

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Step 4.1.1

Substitute $\frac{1}{3}$ for $x$ .

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

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

Multiply $\frac{1}{3} (\frac{1}{3})$ .

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Step 4.1.2.1.1.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 4.1.2.1.1.2

Multiply $3$ by $3$ .

$4 + \frac{1}{9} - \frac{1}{2} \cdot {(\frac{1}{3})}^{2}$

Step 4.1.2.1.2

Apply the product rule to $\frac{1}{3}$ .

$4 + \frac{1}{9} - \frac{1}{2} \cdot \frac{1^{2}}{3^{2}}$

Step 4.1.2.1.3

One to any power is one.

$4 + \frac{1}{9} - \frac{1}{2} \cdot \frac{1}{3^{2}}$

Step 4.1.2.1.4

Raise $3$ to the power of $2$ .

$4 + \frac{1}{9} - \frac{1}{2} \cdot \frac{1}{9}$

Step 4.1.2.1.5

Multiply $- \frac{1}{2} \cdot \frac{1}{9}$ .

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Step 4.1.2.1.5.1

Multiply $\frac{1}{9}$ by $\frac{1}{2}$ .

$4 + \frac{1}{9} - \frac{1}{9 \cdot 2}$

Step 4.1.2.1.5.2

Multiply $9$ by $2$ .

$4 + \frac{1}{9} - \frac{1}{18}$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Write $4$ as a fraction with denominator $1$ .

$\frac{4}{1} + \frac{1}{9} - \frac{1}{18}$

Step 4.1.2.2.2

Multiply $\frac{4}{1}$ by $\frac{18}{18}$ .

$\frac{4}{1} \cdot \frac{18}{18} + \frac{1}{9} - \frac{1}{18}$

Step 4.1.2.2.3

Multiply $\frac{4}{1}$ by $\frac{18}{18}$ .

$\frac{4 \cdot 18}{18} + \frac{1}{9} - \frac{1}{18}$

Step 4.1.2.2.4

Multiply $\frac{1}{9}$ by $\frac{2}{2}$ .

$\frac{4 \cdot 18}{18} + \frac{1}{9} \cdot \frac{2}{2} - \frac{1}{18}$

Step 4.1.2.2.5

Multiply $\frac{1}{9}$ by $\frac{2}{2}$ .

$\frac{4 \cdot 18}{18} + \frac{2}{9 \cdot 2} - \frac{1}{18}$

Step 4.1.2.2.6

Reorder the factors of $9 \cdot 2$ .

$\frac{4 \cdot 18}{18} + \frac{2}{2 \cdot 9} - \frac{1}{18}$

Step 4.1.2.2.7

Multiply $2$ by $9$ .

$\frac{4 \cdot 18}{18} + \frac{2}{18} - \frac{1}{18}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{4 \cdot 18 + 2 - 1}{18}$

Step 4.1.2.4

Simplify the expression.

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Step 4.1.2.4.1

Multiply $4$ by $18$ .

$\frac{72 + 2 - 1}{18}$

Step 4.1.2.4.2

Add $72$ and $2$ .

$\frac{74 - 1}{18}$

Step 4.1.2.4.3

Subtract $1$ from $74$ .

$\frac{73}{18}$

Step 4.2

List all of the points.

$(\frac{1}{3}, \frac{73}{18})$

Step 5