Calculus Examples

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

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 $4 x + 3 x^{2} - x^{3}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x^{3}]$ .

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

Step 1.1.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

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

Step 1.1.2.3

Multiply $4$ by $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

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 $n = 2$ .

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

Step 1.1.3.3

Multiply $2$ by $3$ .

$4 + 6 x + \frac{d}{d x} [- x^{3}]$

Step 1.1.4

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

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Step 1.1.4.1

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

$4 + 6 x - \frac{d}{d x} [x^{3}]$

Step 1.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 + 6 x - (3 x^{2})$

Step 1.1.4.3

Multiply $3$ by $- 1$ .

$4 + 6 x - 3 x^{2}$

Step 1.1.5

Reorder terms.

$f' (x) = - 3 x^{2} + 6 x + 4$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{2} + 6 x + 4$ .

$- 3 x^{2} + 6 x + 4$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 3 x^{2} + 6 x + 4 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 3 x^{2} + 6 x + 4 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.3

Substitute the values $a = - 3$ , $b = 6$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 3 \cdot 4)}}{2 \cdot - 3}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 4}}{2 \cdot - 3}$

Step 2.4.1.2

Multiply $- 4 \cdot - 3 \cdot 4$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot 4}}{2 \cdot - 3}$

Step 2.4.1.2.2

Multiply $12$ by $4$ .

$x = \frac{- 6 \pm \sqrt{36 + 48}}{2 \cdot - 3}$

Step 2.4.1.3

Add $36$ and $48$ .

$x = \frac{- 6 \pm \sqrt{84}}{2 \cdot - 3}$

Step 2.4.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 2.4.1.4.1

Factor $4$ out of $84$ .

$x = \frac{- 6 \pm \sqrt{4 (21)}}{2 \cdot - 3}$

Step 2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot - 3}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{21}}{2 \cdot - 3}$

Step 2.4.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 2 \sqrt{21}}{- 6}$

Step 2.4.3

Simplify $\frac{- 6 \pm 2 \sqrt{21}}{- 6}$ .

$x = \frac{3 \pm \sqrt{21}}{3}$

Step 2.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 4}}{2 \cdot - 3}$

Step 2.5.1.2

Multiply $- 4 \cdot - 3 \cdot 4$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot 4}}{2 \cdot - 3}$

Step 2.5.1.2.2

Multiply $12$ by $4$ .

$x = \frac{- 6 \pm \sqrt{36 + 48}}{2 \cdot - 3}$

Step 2.5.1.3

Add $36$ and $48$ .

$x = \frac{- 6 \pm \sqrt{84}}{2 \cdot - 3}$

Step 2.5.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 2.5.1.4.1

Factor $4$ out of $84$ .

$x = \frac{- 6 \pm \sqrt{4 (21)}}{2 \cdot - 3}$

Step 2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot - 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{21}}{2 \cdot - 3}$

Step 2.5.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 2 \sqrt{21}}{- 6}$

Step 2.5.3

Simplify $\frac{- 6 \pm 2 \sqrt{21}}{- 6}$ .

$x = \frac{3 \pm \sqrt{21}}{3}$

Step 2.5.4

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{21}}{3}$

Step 2.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 4}}{2 \cdot - 3}$

Step 2.6.1.2

Multiply $- 4 \cdot - 3 \cdot 4$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot 4}}{2 \cdot - 3}$

Step 2.6.1.2.2

Multiply $12$ by $4$ .

$x = \frac{- 6 \pm \sqrt{36 + 48}}{2 \cdot - 3}$

Step 2.6.1.3

Add $36$ and $48$ .

$x = \frac{- 6 \pm \sqrt{84}}{2 \cdot - 3}$

Step 2.6.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 2.6.1.4.1

Factor $4$ out of $84$ .

$x = \frac{- 6 \pm \sqrt{4 (21)}}{2 \cdot - 3}$

Step 2.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot - 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{21}}{2 \cdot - 3}$

Step 2.6.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 2 \sqrt{21}}{- 6}$

Step 2.6.3

Simplify $\frac{- 6 \pm 2 \sqrt{21}}{- 6}$ .

$x = \frac{3 \pm \sqrt{21}}{3}$

Step 2.6.4

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{21}}{3}$

Step 2.7

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{21}}{3}, \frac{3 - \sqrt{21}}{3}$

Step 3

The values which make the derivative equal to $0$ are $\frac{3 + \sqrt{21}}{3}, \frac{3 - \sqrt{21}}{3}$ .

$\frac{3 + \sqrt{21}}{3}, \frac{3 - \sqrt{21}}{3}$

Step 4

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

$(- \infty, \frac{3 - \sqrt{21}}{3}) \cup (\frac{3 - \sqrt{21}}{3}, \frac{3 + \sqrt{21}}{3}) \cup (\frac{3 + \sqrt{21}}{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{3 - \sqrt{21}}{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.52752525$ in the expression.

$f' (- 1.52752525) = - 3 {(- 1.52752525)}^{2} + 6 (- 1.52752525) + 4$

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 $- 1.52752525$ to the power of $2$ .

$f' (- 1.52752525) = - 3 \cdot 2.33333338 + 6 (- 1.52752525) + 4$

Step 5.2.1.2

Multiply $- 3$ by $2.33333338$ .

$f' (- 1.52752525) = - 7.00000016 + 6 (- 1.52752525) + 4$

Step 5.2.1.3

Multiply $6$ by $- 1.52752525$ .

$f' (- 1.52752525) = - 7.00000016 - 9.1651515 + 4$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $9.1651515$ from $- 7.00000016$ .

$f' (- 1.52752525) = - 16.16515166 + 4$

Step 5.2.2.2

Add $- 16.16515166$ and $4$ .

$f' (- 1.52752525) = - 12.16515166$

Step 5.2.3

The final answer is $- 12.16515166$ .

$- 12.16515166$

Step 5.3

At $x = - 1.52752525$ the derivative is $- 12.16515166$ . Since this is negative, the function is decreasing on $(- \infty, \frac{3 - \sqrt{21}}{3})$ .

Decreasing on $(- \infty, \frac{3 - \sqrt{21}}{3})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 0.52752525, \frac{3 + \sqrt{21}}{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.99999997$ in the expression.

$f' (0.99999997) = - 3 {(0.99999997)}^{2} + 6 (0.99999997) + 4$

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 $0.99999997$ to the power of $2$ .

$f' (0.99999997) = - 3 \cdot 0.99999995 + 6 (0.99999997) + 4$

Step 6.2.1.2

Multiply $- 3$ by $0.99999995$ .

$f' (0.99999997) = - 2.99999985 + 6 (0.99999997) + 4$

Step 6.2.1.3

Multiply $6$ by $0.99999997$ .

$f' (0.99999997) = - 2.99999985 + 5.99999985 + 4$

Step 6.2.2

Simplify by adding numbers.

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Step 6.2.2.1

Add $- 2.99999985$ and $5.99999985$ .

$f' (0.99999997) = 3 + 4$

Step 6.2.2.2

Add $3$ and $4$ .

$f' (0.99999997) = 7$

Step 6.2.3

The final answer is $7$ .

$7$

Step 6.3

At $x = 0.99999997$ the derivative is $7$ . Since this is positive, the function is increasing on $(- 0.52752525, \frac{3 + \sqrt{21}}{3})$ .

Increasing on $(\frac{3 - \sqrt{21}}{3}, \frac{3 + \sqrt{21}}{3})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(\frac{3 + \sqrt{21}}{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.5275252$ in the expression.

$f' (3.5275252) = - 3 {(3.5275252)}^{2} + 6 (3.5275252) + 4$

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 $3.5275252$ to the power of $2$ .

$f' (3.5275252) = - 3 \cdot 12.44343403 + 6 (3.5275252) + 4$

Step 7.2.1.2

Multiply $- 3$ by $12.44343403$ .

$f' (3.5275252) = - 37.3303021 + 6 (3.5275252) + 4$

Step 7.2.1.3

Multiply $6$ by $3.5275252$ .

$f' (3.5275252) = - 37.3303021 + 21.1651512 + 4$

Step 7.2.2

Simplify by adding numbers.

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Step 7.2.2.1

Add $- 37.3303021$ and $21.1651512$ .

$f' (3.5275252) = - 16.1651509 + 4$

Step 7.2.2.2

Add $- 16.1651509$ and $4$ .

$f' (3.5275252) = - 12.1651509$

Step 7.2.3

The final answer is $- 12.1651509$ .

$- 12.1651509$

Step 7.3

At $x = 3.5275252$ the derivative is $- 12.1651509$ . Since this is negative, the function is decreasing on $(\frac{3 + \sqrt{21}}{3}, \infty)$ .

Decreasing on $(\frac{3 + \sqrt{21}}{3}, \infty)$ since $f' (x) < 0$

Step 8

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

Increasing on: $(\frac{3 - \sqrt{21}}{3}, \frac{3 + \sqrt{21}}{3})$

Decreasing on: $(- \infty, \frac{3 - \sqrt{21}}{3}), (\frac{3 + \sqrt{21}}{3}, \infty)$

Step 9