Calculus Examples

$f (x) = x^{4} - 12 x^{2} + 36$

Find the derivative.

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Differentiate.

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By the Sum Rule, the derivative of $x^{4} - 12 x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [36]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [36]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [36]$

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]$

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

$4 x^{3} - 12 (2 x) + \frac{d}{d x} [36]$

Multiply $2$ by $- 12$ .

$4 x^{3} - 24 x + \frac{d}{d x} [36]$

Differentiate using the Constant Rule.

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Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$4 x^{3} - 24 x + 0$

Add $4 x^{3} - 24 x$ and $0$ .

$4 x^{3} - 24 x$

Set the derivative equal to $0$ .

$4 x^{3} - 24 x = 0$

Solve for $x$ .

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Factor $4 x$ out of $4 x^{3} - 24 x$ .

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Factor $4 x$ out of $4 x^{3}$ .

$4 x (x^{2}) - 24 x = 0$

Factor $4 x$ out of $- 24 x$ .

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

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 6)$ .

$4 x (x^{2} - 6) = 0$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = 0$ by $4$ .

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Divide $0$ by $4$ .

$x = 0$

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Set the factor equal to $0$ .

$x^{2} - 6 = 0$

Add $6$ to both sides of the equation.

$x^{2} = 6$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{6}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{6}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{6}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{6}, - \sqrt{6}$

The solution is the result of $4 x = 0$ and $x^{2} - 6 = 0$ .

$x = 0, \sqrt{6}, - \sqrt{6}$

The values which make the derivative equal to $0$ are $0, \sqrt{6}, - \sqrt{6}$ .

$0, \sqrt{6}, - \sqrt{6}$

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

$(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, 0) \cup (0, \sqrt{6}) \cup (\sqrt{6}, \infty)$

Substitute a value from the interval $(- \infty, - \sqrt{6})$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $- 3.44949$ in the expression.

$f' (- 3.44949) = 4 {(- 3.44949)}^{3} - 24 \cdot - 3.44949$

Simplify the result.

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Simplify each term.

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Raise $- 3.44949$ to the power of $3$ .

$f' (- 3.44949) = 4 \cdot - 41.04541686 - 24 \cdot - 3.44949$

Multiply $4$ by $- 41.04541686$ .

$f' (- 3.44949) = - 164.18166746 - 24 \cdot - 3.44949$

Multiply $- 24$ by $- 3.44949$ .

$f' (- 3.44949) = - 164.18166746 + 82.78776$

Add $- 164.18166746$ and $82.78776$ .

$f' (- 3.44949) = - 81.39390746$

The final answer is $- 81.39390746$ .

$- 81.39390746$

At $x = - 3.44949$ the derivative is $- 81.39390746$ . Since this is negative, the function is decreasing on $(- \infty, - \sqrt{6})$ .

Decreasing on $(- \infty, - \sqrt{6})$ since $f' (x) < 0$

Substitute a value from the interval $(- 2.44949, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $- 1.224745$ in the expression.

$f' (- 1.224745) = 4 {(- 1.224745)}^{3} - 24 \cdot - 1.224745$

Simplify the result.

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Simplify each term.

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Raise $- 1.224745$ to the power of $3$ .

$f' (- 1.224745) = 4 \cdot - 1.83711788 - 24 \cdot - 1.224745$

Multiply $4$ by $- 1.83711788$ .

$f' (- 1.224745) = - 7.34847154 - 24 \cdot - 1.224745$

Multiply $- 24$ by $- 1.224745$ .

$f' (- 1.224745) = - 7.34847154 + 29.39388$

Add $- 7.34847154$ and $29.39388$ .

$f' (- 1.224745) = 22.04540845$

The final answer is $22.04540845$ .

$22.04540845$

At $x = - 1.224745$ the derivative is $22.04540845$ . Since this is positive, the function is increasing on $(- 2.44949, 0)$ .

Increasing on $(- \sqrt{6}, 0)$ since $f' (x) > 0$

Substitute a value from the interval $(0, \sqrt{6})$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $1.224745$ in the expression.

$f' (1.224745) = 4 {(1.224745)}^{3} - 24 \cdot 1.224745$

Simplify the result.

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Simplify each term.

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Raise $1.224745$ to the power of $3$ .

$f' (1.224745) = 4 \cdot 1.83711788 - 24 \cdot 1.224745$

Multiply $4$ by $1.83711788$ .

$f' (1.224745) = 7.34847154 - 24 \cdot 1.224745$

Multiply $- 24$ by $1.224745$ .

$f' (1.224745) = 7.34847154 - 29.39388$

Subtract $29.39388$ from $7.34847154$ .

$f' (1.224745) = - 22.04540845$

The final answer is $- 22.04540845$ .

$- 22.04540845$

At $x = 1.224745$ the derivative is $- 22.04540845$ . Since this is negative, the function is decreasing on $(0, \sqrt{6})$ .

Decreasing on $(0, \sqrt{6})$ since $f' (x) < 0$

Substitute a value from the interval $(\sqrt{6}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $3.44949$ in the expression.

$f' (3.44949) = 4 {(3.44949)}^{3} - 24 \cdot 3.44949$

Simplify the result.

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Simplify each term.

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Raise $3.44949$ to the power of $3$ .

$f' (3.44949) = 4 \cdot 41.04541686 - 24 \cdot 3.44949$

Multiply $4$ by $41.04541686$ .

$f' (3.44949) = 164.18166746 - 24 \cdot 3.44949$

Multiply $- 24$ by $3.44949$ .

$f' (3.44949) = 164.18166746 - 82.78776$

Subtract $82.78776$ from $164.18166746$ .

$f' (3.44949) = 81.39390746$

The final answer is $81.39390746$ .

$81.39390746$

At $x = 3.44949$ the derivative is $81.39390746$ . Since this is positive, the function is increasing on $(\sqrt{6}, \infty)$ .

Increasing on $(\sqrt{6}, \infty)$ since $f' (x) > 0$

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

Increasing on: $(- \sqrt{6}, 0), (\sqrt{6}, \infty)$

Decreasing on: $(- \infty, - \sqrt{6}), (0, \sqrt{6})$

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