Calculus Examples

$f (x) = 2 x + 3$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

$\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$

Step 1.1.2

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

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Step 1.1.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.2.2

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

$2 \cdot 1 + \frac{d}{d x} [3]$

Step 1.1.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [3]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 + 0$

Step 1.1.3.2

Add $2$ and $0$ .

$f' (x) = 2$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2$ .

$2$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 = 0$

Step 2.2

Since $2 \neq 0$ , there are no solutions.

No solution

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found