Calculus Examples

$f (x) = 4 x^{3} + 12 x - 2$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4 x^{3} + 12 x - 2$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 2]$ .

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

Step 1.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $3$ by $4$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$12 x^{2} + 12 \cdot 1 + \frac{d}{d x} [- 2]$

Step 1.3.3

Multiply $12$ by $1$ .

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

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$12 x^{2} + 12 + 0$

Step 1.4.2

Add $12 x^{2} + 12$ and $0$ .

$12 x^{2} + 12$

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$f'' (x) = 12 (2 x) + \frac{d}{d x} (12)$

Step 2.2.3

Multiply $2$ by $12$ .

$f'' (x) = 24 x + \frac{d}{d x} (12)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$f'' (x) = 24 x + 0$

Step 2.3.2

Add $24 x$ and $0$ .

$f'' (x) = 24 x$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$12 x^{2} + 12 = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6