Calculus Examples

$f (x) = 4 x - 2$ , $(1, 3)$

Find the derivative.

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Find the first derivative.

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

$f' (x) = \frac{d}{d x} (4 x) + \frac{d}{d x} (- 2)$

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

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

$f' (x) = 4 \frac{d}{d x} (x) + \frac{d}{d x} (- 2)$

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

$f' (x) = 4 \cdot 1 + \frac{d}{d x} (- 2)$

Multiply $4$ by $1$ .

$f' (x) = 4 + \frac{d}{d x} (- 2)$

Differentiate using the Constant Rule.

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

$f' (x) = 4 + 0$

Add $4$ and $0$ .

$f' (x) = 4$

The first derivative of $f (x)$ with respect to $x$ is $4$ .

$4$

Find if the derivative is continuous on $(1, 3)$ .

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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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

$f' (x)$ is continuous on $(1, 3)$ .

The function is continuous.

The function is differentiable on $(1, 3)$ because the derivative is continuous on $(1, 3)$ .

The function is differentiable.

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