Calculus Examples

$2 x - 4$

Step 1

Write $2 x - 4$ as a function.

$f (x) = 2 x - 4$

Step 2

Find the first derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $1$ .

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

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.3.2

Add $2$ and $0$ .

$2$

Step 3

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

$f'' (x) = 0$

Step 4

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

$2 = 0$

Step 5

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

No Local Extrema

Step 6

No Local Extrema

Step 7