Calculus Examples

$f (x) = 3 x^{3} + x + 3$ , $(5, 7)$

Step 1

Find the 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 $3 x^{3} + x + 3$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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 $n = 3$ .

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

Step 1.1.2.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$9 x^{2} + 1 + \frac{d}{d x} [3]$

Step 1.1.3.2

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

$9 x^{2} + 1 + 0$

Step 1.1.3.3

Add $9 x^{2} + 1$ and $0$ .

$f' (x) = 9 x^{2} + 1$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $9 x^{2} + 1$ .

$9 x^{2} + 1$

Step 2

Find if the derivative is continuous on $(5, 7)$ .

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Step 2.1

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:

${x | x \in ℝ}$

Step 2.2

$f' (x)$ is continuous on $(5, 7)$ .

The function is continuous.

Step 3

The function is differentiable on $(5, 7)$ because the derivative is continuous on $(5, 7)$ .

The function is differentiable.

Step 4

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