Calculus Examples

$f (x) = 2 x^{2} \cdot \frac{4000}{x} + 13$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{2} \cdot \frac{4000}{x} + 13$ with respect to $x$ is $\frac{d}{d x} [2 x^{2} \cdot \frac{4000}{x}] + \frac{d}{d x} [13]$ .

$\frac{d}{d x} [2 x^{2} \cdot \frac{4000}{x}] + \frac{d}{d x} [13]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x^{2} \cdot \frac{4000}{x}]$ .

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Step 1.2.1

Combine $2$ and $\frac{4000}{x}$ .

$\frac{d}{d x} [x^{2} \cdot \frac{2 \cdot 4000}{x}] + \frac{d}{d x} [13]$

Step 1.2.2

Multiply $2$ by $4000$ .

$\frac{d}{d x} [x^{2} \cdot \frac{8000}{x}] + \frac{d}{d x} [13]$

Step 1.2.3

Combine $x^{2}$ and $\frac{8000}{x}$ .

$\frac{d}{d x} [\frac{x^{2} \cdot 8000}{x}] + \frac{d}{d x} [13]$

Step 1.2.4

Move $8000$ to the left of $x^{2}$ .

$\frac{d}{d x} [\frac{8000 \cdot x^{2}}{x}] + \frac{d}{d x} [13]$

Step 1.2.5

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.2.5.1

Factor $x$ out of $8000 x^{2}$ .

$\frac{d}{d x} [\frac{x (8000 x)}{x}] + \frac{d}{d x} [13]$

Step 1.2.5.2

Cancel the common factors.

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Step 1.2.5.2.1

Raise $x$ to the power of $1$ .

$\frac{d}{d x} [\frac{x (8000 x)}{x^{1}}] + \frac{d}{d x} [13]$

Step 1.2.5.2.2

Factor $x$ out of $x^{1}$ .

$\frac{d}{d x} [\frac{x (8000 x)}{x \cdot 1}] + \frac{d}{d x} [13]$

Step 1.2.5.2.3

Cancel the common factor.

$\frac{d}{d x} [\frac{x (8000 x)}{x \cdot 1}] + \frac{d}{d x} [13]$

Step 1.2.5.2.4

Rewrite the expression.

$\frac{d}{d x} [\frac{8000 x}{1}] + \frac{d}{d x} [13]$

Step 1.2.5.2.5

Divide $8000 x$ by $1$ .

$\frac{d}{d x} [8000 x] + \frac{d}{d x} [13]$

Step 1.2.6

Since $8000$ is constant with respect to $x$ , the derivative of $8000 x$ with respect to $x$ is $8000 \frac{d}{d x} [x]$ .

$8000 \frac{d}{d x} [x] + \frac{d}{d x} [13]$

Step 1.2.7

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

$8000 \cdot 1 + \frac{d}{d x} [13]$

Step 1.2.8

Multiply $8000$ by $1$ .

$8000 + \frac{d}{d x} [13]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $13$ is constant with respect to $x$ , the derivative of $13$ with respect to $x$ is $0$ .

$8000 + 0$

Step 1.3.2

Add $8000$ and $0$ .

$8000$

Step 2

Since $8000$ is constant with respect to $x$ , the derivative of $8000$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 3

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

$8000 = 0$

Step 4

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

No Local Extrema

Step 5

No Local Extrema

Step 6