Calculus Examples

$6 x - 12$

Step 1

Write $6 x - 12$ as a function.

$f (x) = 6 x - 12$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $6 x - 12$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 12]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 12]$

Step 2.1.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 12]$

Step 2.1.2.2

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

$6 \cdot 1 + \frac{d}{d x} [- 12]$

Step 2.1.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 12]$

Step 2.1.3

Differentiate using the Constant Rule.

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Step 2.1.3.1

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

$6 + 0$

Step 2.1.3.2

Add $6$ and $0$ .

$f' (x) = 6$

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $0$ .

$0$

Step 3

Set the second derivative equal to $0$ then solve the equation $0 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$0 = 0$

Step 3.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 4

There are no inflection points in a straight line, $f (x) = 6 x - 12$ .

No Inflection Points