Calculus Examples

$x^{3} + 12 x^{2} + 36 x + 8$

Step 1

Find the first derivative.

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Find the first derivative.

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Differentiate.

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By the Sum Rule, the derivative of $x^{3} + 12 x^{2} + 36 x + 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [36 x] + \frac{d}{d x} [8]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (36 x) + \frac{d}{d x} (8)$

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

$f' (x) = 3 x^{2} + \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (36 x) + \frac{d}{d x} (8)$

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

$f' (x) = 3 x^{2} + 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (36 x) + \frac{d}{d x} (8)$

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

$f' (x) = 3 x^{2} + 12 (2 x) + \frac{d}{d x} (36 x) + \frac{d}{d x} (8)$

Multiply $2$ by $12$ .

$f' (x) = 3 x^{2} + 24 x + \frac{d}{d x} (36 x) + \frac{d}{d x} (8)$

Evaluate $\frac{d}{d x} [36 x]$ .

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Since $36$ is constant with respect to $x$ , the derivative of $36 x$ with respect to $x$ is $36 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} + 24 x + 36 \frac{d}{d x} (x) + \frac{d}{d x} (8)$

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

$f' (x) = 3 x^{2} + 24 x + 36 \cdot 1 + \frac{d}{d x} (8)$

Multiply $36$ by $1$ .

$f' (x) = 3 x^{2} + 24 x + 36 + \frac{d}{d x} (8)$

Differentiate using the Constant Rule.

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Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f' (x) = 3 x^{2} + 24 x + 36 + 0$

Add $3 x^{2} + 24 x + 36$ and $0$ .

$f' (x) = 3 x^{2} + 24 x + 36$

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 24 x + 36$ .

$3 x^{2} + 24 x + 36$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + 24 x + 36 = 0$ .

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Set the first derivative equal to $0$ .

$3 x^{2} + 24 x + 36 = 0$

Factor the left side of the equation.

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Factor $3$ out of $3 x^{2} + 24 x + 36$ .

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Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 24 x + 36 = 0$

Factor $3$ out of $24 x$ .

$3 (x^{2}) + 3 (8 x) + 36 = 0$

Factor $3$ out of $36$ .

$3 x^{2} + 3 (8 x) + 3 \cdot 12 = 0$

Factor $3$ out of $3 x^{2} + 3 (8 x)$ .

$3 (x^{2} + 8 x) + 3 \cdot 12 = 0$

Factor $3$ out of $3 (x^{2} + 8 x) + 3 \cdot 12$ .

$3 (x^{2} + 8 x + 12) = 0$

Factor.

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Factor $x^{2} + 8 x + 12$ using the AC method.

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Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $8$ .

$2, 6$

Write the factored form using these integers.

$3 ((x + 2) (x + 6)) = 0$

Remove unnecessary parentheses.

$3 (x + 2) (x + 6) = 0$

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

$x + 2 = 0$

$x + 6 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

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Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x + 6$ equal to $0$ and solve for $x$ .

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Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $3 (x + 2) (x + 6) = 0$ true.

$x = - 2, - 6$

Step 3

Find the values where the derivative is undefined.

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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 $x^{3} + 12 x^{2} + 36 x + 8$ at each $x$ value where the derivative is $0$ or undefined.

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Evaluate at $x = - 2$ .

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Substitute $- 2$ for $x$ .

${(- 2)}^{3} + 12 {(- 2)}^{2} + 36 (- 2) + 8$

Simplify.

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Simplify each term.

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

$- 8 + 12 {(- 2)}^{2} + 36 (- 2) + 8$

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

$- 8 + 12 \cdot 4 + 36 (- 2) + 8$

Multiply $12$ by $4$ .

$- 8 + 48 + 36 (- 2) + 8$

Multiply $36$ by $- 2$ .

$- 8 + 48 - 72 + 8$

Simplify by adding and subtracting.

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Add $- 8$ and $48$ .

$40 - 72 + 8$

Subtract $72$ from $40$ .

$- 32 + 8$

Add $- 32$ and $8$ .

$- 24$

Evaluate at $x = - 6$ .

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Substitute $- 6$ for $x$ .

${(- 6)}^{3} + 12 {(- 6)}^{2} + 36 (- 6) + 8$

Simplify.

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Simplify each term.

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Raise $- 6$ to the power of $3$ .

$- 216 + 12 {(- 6)}^{2} + 36 (- 6) + 8$

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

$- 216 + 12 \cdot 36 + 36 (- 6) + 8$

Multiply $12$ by $36$ .

$- 216 + 432 + 36 (- 6) + 8$

Multiply $36$ by $- 6$ .

$- 216 + 432 - 216 + 8$

Simplify by adding and subtracting.

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Add $- 216$ and $432$ .

$216 - 216 + 8$

Subtract $216$ from $216$ .

$0 + 8$

Add $0$ and $8$ .

$8$

List all of the points.

$(- 2, - 24), (- 6, 8)$

Step 5