Calculus Examples

$f (x) = \frac{2 x}{3} + 3 x$ , $- 1 < x < - 0$

Step 1

Find the derivative of $f (x) = \frac{2 x}{3} + 3 x$ .

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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 $\frac{2 x}{3} + 3 x$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x}{3}] + \frac{d}{d x} [3 x]$ .

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

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{2 x}{3}]$ .

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Step 1.1.2.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x}{3}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

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

Step 1.1.2.3

Multiply $\frac{2}{3}$ by $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

$\frac{2}{3} + 3 \cdot 1$

Step 1.1.3.3

Multiply $3$ by $1$ .

$\frac{2}{3} + 3$

Step 1.1.4

Combine terms.

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Step 1.1.4.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} + 3 \cdot \frac{3}{3}$

Step 1.1.4.2

Combine $3$ and $\frac{3}{3}$ .

$\frac{2}{3} + \frac{3 \cdot 3}{3}$

Step 1.1.4.3

Combine the numerators over the common denominator.

$\frac{2 + 3 \cdot 3}{3}$

Step 1.1.4.4

Simplify the numerator.

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Step 1.1.4.4.1

Multiply $3$ by $3$ .

$\frac{2 + 9}{3}$

Step 1.1.4.4.2

Add $2$ and $9$ .

$f' (x) = \frac{11}{3}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{11}{3}$ .

$\frac{11}{3}$

Step 2

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 3

$f' (x)$ is continuous on $[- 1, 0]$ .

$f' (x)$ is continuous

Step 4

The average value of function $f'$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 5

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{0 + 1} (\int_{- 1}^{0} \frac{11}{3} d x)$

Step 6

Apply the constant rule.

$A (x) = \frac{1}{0 + 1} (\frac{11}{3} x]_{- 1}^{0})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{11}{3} x$ at $0$ and at $- 1$ .

$A (x) = \frac{1}{0 + 1} ((\frac{11}{3} \cdot 0) - \frac{11}{3} \cdot - 1)$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{11}{3}$ by $0$ .

$A (x) = \frac{1}{0 + 1} (0 - \frac{11}{3} \cdot - 1)$

Step 7.2.2

Multiply $- 1$ by $- 1$ .

$A (x) = \frac{1}{0 + 1} (0 + 1 (\frac{11}{3}))$

Step 7.2.3

Multiply $\frac{11}{3}$ by $1$ .

$A (x) = \frac{1}{0 + 1} (0 + \frac{11}{3})$

Step 7.2.4

Add $0$ and $\frac{11}{3}$ .

$A (x) = \frac{1}{0 + 1} (\frac{11}{3})$

Step 8

Add $0$ and $1$ .

$A (x) = \frac{1}{1} \cdot \frac{11}{3}$

Step 9

Cancel the common factor of $1$ .

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Step 9.1

Cancel the common factor.

$A (x) = \frac{1}{1} \cdot \frac{11}{3}$

Step 9.2

Rewrite the expression.

$A (x) = 1 \cdot \frac{11}{3}$

Step 10

Multiply $\frac{11}{3}$ by $1$ .

$A (x) = \frac{11}{3}$

Step 11