Calculus Examples

$g (x) = x^{2} \cdot \frac{2}{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

Simplify terms.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Cancel the common factors.

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Step 1.1.3.2.1

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

$\frac{d}{d x} [\frac{x (2 x)}{x^{1}}]$

Step 1.1.3.2.2

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

$\frac{d}{d x} [\frac{x (2 x)}{x \cdot 1}]$

Step 1.1.3.2.3

Cancel the common factor.

$\frac{d}{d x} [\frac{x (2 x)}{x \cdot 1}]$

Step 1.1.3.2.4

Rewrite the expression.

$\frac{d}{d x} [\frac{2 x}{1}]$

Step 1.1.3.2.5

Divide $2 x$ by $1$ .

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

Step 1.2

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

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

Step 1.3

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

$2 \cdot 1$

Step 1.4

Multiply $2$ by $1$ .

$2$

Step 2

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

$g'' (x) = 0$

Step 3

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

$2 = 0$

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