Calculus Examples

$f (x) = 3 x^{2} - 8 x^{3}$

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 $3 x^{2} - 8 x^{3}$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x^{3}]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x^{3}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.1.2.1

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

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x^{3}]$

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 $n = 2$ .

$3 (2 x) + \frac{d}{d x} [- 8 x^{3}]$

Step 1.1.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 8 x^{3}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 1.1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$6 x - 8 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$6 x - 8 (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $- 8$ .

$6 x - 24 x^{2}$

Step 1.1.4

Reorder terms.

$f' (x) = - 24 x^{2} + 6 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 24 x^{2} + 6 x$ .

$- 24 x^{2} + 6 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 24 x^{2} + 6 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 24 x^{2} + 6 x = 0$

Step 2.2

Factor $- 6 x$ out of $- 24 x^{2} + 6 x$ .

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Step 2.2.1

Factor $- 6 x$ out of $- 24 x^{2}$ .

$- 6 x (4 x) + 6 x = 0$

Step 2.2.2

Factor $- 6 x$ out of $6 x$ .

$- 6 x (4 x) - 6 x \cdot - 1 = 0$

Step 2.2.3

Factor $- 6 x$ out of $- 6 x (4 x) - 6 x (- 1)$ .

$- 6 x (4 x - 1) = 0$

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 $0$ .

$x = 0$

$4 x - 1 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 2.5.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 2.5.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 2.5.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 2.5.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 2.6

The final solution is all the values that make $- 6 x (4 x - 1) = 0$ true.

$x = 0, \frac{1}{4}$

Step 3

The values which make the derivative equal to $0$ are $0, \frac{1}{4}$ .

$0, \frac{1}{4}$

Step 4

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

$(- \infty, 0) \cup (0, \frac{1}{4}) \cup (\frac{1}{4}, \infty)$

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.

$f' (- 1) = - 24 {(- 1)}^{2} + 6 (- 1)$

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 $2$ .

$f' (- 1) = - 24 \cdot 1 + 6 (- 1)$

Step 5.2.1.2

Multiply $- 24$ by $1$ .

$f' (- 1) = - 24 + 6 (- 1)$

Step 5.2.1.3

Multiply $6$ by $- 1$ .

$f' (- 1) = - 24 - 6$

Step 5.2.2

Subtract $6$ from $- 24$ .

$f' (- 1) = - 30$

Step 5.2.3

The final answer is $- 30$ .

$- 30$

Step 5.3

At $x = - 1$ the derivative is $- 30$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(0, \frac{1}{4})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.125$ in the expression.

$f' (0.125) = - 24 {(0.125)}^{2} + 6 (0.125)$

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

Raise $0.125$ to the power of $2$ .

$f' (0.125) = - 24 \cdot 0.015625 + 6 (0.125)$

Step 6.2.1.2

Multiply $- 24$ by $0.015625$ .

$f' (0.125) = - 0.375 + 6 (0.125)$

Step 6.2.1.3

Multiply $6$ by $0.125$ .

$f' (0.125) = - 0.375 + 0.75$

Step 6.2.2

Add $- 0.375$ and $0.75$ .

$f' (0.125) = 0.375$

Step 6.2.3

The final answer is $0.375$ .

$0.375$

Step 6.3

At $x = 0.125$ the derivative is $0.375$ . Since this is positive, the function is increasing on $(0, \frac{1}{4})$ .

Increasing on $(0, \frac{1}{4})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(\frac{1}{4}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.25$ in the expression.

$f' (1.25) = - 24 {(1.25)}^{2} + 6 (1.25)$

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 $1.25$ to the power of $2$ .

$f' (1.25) = - 24 \cdot 1.5625 + 6 (1.25)$

Step 7.2.1.2

Multiply $- 24$ by $1.5625$ .

$f' (1.25) = - 37.5 + 6 (1.25)$

Step 7.2.1.3

Multiply $6$ by $1.25$ .

$f' (1.25) = - 37.5 + 7.5$

Step 7.2.2

Add $- 37.5$ and $7.5$ .

$f' (1.25) = - 30$

Step 7.2.3

The final answer is $- 30$ .

$- 30$

Step 7.3

At $x = 1.25$ the derivative is $- 30$ . Since this is negative, the function is decreasing on $(\frac{1}{4}, \infty)$ .

Decreasing on $(\frac{1}{4}, \infty)$ since $f' (x) < 0$

Step 8

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

Increasing on: $(0, \frac{1}{4})$

Decreasing on: $(- \infty, 0), (\frac{1}{4}, \infty)$

Step 9