Calculus Examples

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

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

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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 $n = 3$ .

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

Step 1.1.2.3

Combine $3$ and $\frac{1}{3}$ .

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

Step 1.1.2.4

Combine $\frac{3}{3}$ and $x^{2}$ .

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

Step 1.1.2.5

Cancel the common factor of $3$ .

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Step 1.1.2.5.1

Cancel the common factor.

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

Step 1.1.2.5.2

Divide $x^{2}$ by $1$ .

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

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

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

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 $n = 2$ .

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

Step 1.1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.1.3.4

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

$x^{2} + \frac{- 2}{2} x$

Step 1.1.3.5

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

$x^{2} + \frac{- 2 x}{2}$

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 $- 2 x$ .

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

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 $2$ .

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

Step 1.1.3.6.2.2

Cancel the common factor.

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

Step 1.1.3.6.2.3

Rewrite the expression.

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

Step 1.1.3.6.2.4

Divide $- x$ by $1$ .

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

Step 1.2

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

$x^{2} - x$

Step 2

Set the first derivative equal to $0$ then solve the equation $x^{2} - x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$x^{2} - x = 0$

Step 2.2

Factor $x$ out of $x^{2} - x$ .

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Step 2.2.1

Factor $x$ out of $x^{2}$ .

$x \cdot x - x = 0$

Step 2.2.2

Factor $x$ out of $- x$ .

$x \cdot x + x \cdot - 1 = 0$

Step 2.2.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$x (x - 1) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 1 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

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

$x - 1 = 0$

Step 2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.6

The final solution is all the values that make $x (x - 1) = 0$ true.

$x = 0, 1$

Step 3

The values which make the derivative equal to $0$ are $0, 1$ .

$0, 1$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 1) \cup (1, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f' (- 1) = 1 - (- 1)$

Step 5.2.1.2

Multiply $- 1$ by $- 1$ .

$f' (- 1) = 1 + 1$

Step 5.2.2

Add $1$ and $1$ .

$f' (- 1) = 2$

Step 5.2.3

The final answer is $2$ .

$2$

Step 5.3

At $x = - 1$ the derivative is $2$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(0, 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 6.2.1.2

One to any power is one.

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

Step 6.2.1.3

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

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

Step 6.2.2

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

$f' (\frac{1}{2}) = \frac{1}{4} - \frac{1}{2} \cdot \frac{2}{2}$

Step 6.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.1

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

$f' (\frac{1}{2}) = \frac{1}{4} - \frac{2}{2 \cdot 2}$

Step 6.2.3.2

Multiply $2$ by $2$ .

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

Step 6.2.4

Combine the numerators over the common denominator.

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

Step 6.2.5

Subtract $2$ from $1$ .

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

Step 6.2.6

Move the negative in front of the fraction.

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

Step 6.2.7

The final answer is $- \frac{1}{4}$ .

$- \frac{1}{4}$

Step 6.3

At $x = \frac{1}{2}$ the derivative is $- \frac{1}{4}$ . Since this is negative, the function is decreasing on $(0, 1)$ .

Decreasing on $(0, 1)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = {(2)}^{2} - (2)$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

$f' (2) = 4 - (2)$

Step 7.2.1.2

Multiply $- 1$ by $2$ .

$f' (2) = 4 - 2$

Step 7.2.2

Subtract $2$ from $4$ .

$f' (2) = 2$

Step 7.2.3

The final answer is $2$ .

$2$

Step 7.3

At $x = 2$ the derivative is $2$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, 0), (1, \infty)$

Decreasing on: $(0, 1)$

Step 9