Calculus Examples

$(3 x^{2}) (x^{\frac{1}{2}})$

Step 1

Multiply $x^{2}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1

Move $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [3 (x^{\frac{1}{2}} x^{2})]$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $2$ by $2$ .

$\frac{d}{d x} [3 x^{\frac{1 + 4}{2}}]$

Step 1.6.2

Add $1$ and $4$ .

$\frac{d}{d x} [3 x^{\frac{5}{2}}]$

Step 2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{5}{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$3 \frac{d}{d x} [x^{\frac{5}{2}}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$3 (\frac{5}{2} x^{\frac{5}{2} - 1})$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

$3 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 6

Combine the numerators over the common denominator.

$3 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$3 (\frac{5}{2} x^{\frac{5 - 2}{2}})$

Step 7.2

Subtract $2$ from $5$ .

$3 (\frac{5}{2} x^{\frac{3}{2}})$

Step 8

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$3 \frac{5 x^{\frac{3}{2}}}{2}$

Step 9

Combine $3$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$\frac{3 (5 x^{\frac{3}{2}})}{2}$

Step 10

Multiply $5$ by $3$ .

$\frac{15 x^{\frac{3}{2}}}{2}$