Calculus Examples

$y = x^{\frac{5}{2}} + 2 \ln (x)$

Step 1

By the Sum Rule, the derivative of $x^{\frac{5}{2}} + 2 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [2 \ln (x)]$ .

$\frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [2 \ln (x)]$

Step 2

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

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Step 2.1

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

$\frac{5}{2} x^{\frac{5}{2} - 1} + \frac{d}{d x} [2 \ln (x)]$

Step 2.2

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

$\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [2 \ln (x)]$

Step 2.3

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

$\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [2 \ln (x)]$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}} + \frac{d}{d x} [2 \ln (x)]$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $2$ .

$\frac{5}{2} x^{\frac{5 - 2}{2}} + \frac{d}{d x} [2 \ln (x)]$

Step 2.5.2

Subtract $2$ from $5$ .

$\frac{5}{2} x^{\frac{3}{2}} + \frac{d}{d x} [2 \ln (x)]$

Step 3

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

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Step 3.1

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

$\frac{5}{2} x^{\frac{3}{2}} + 2 \frac{d}{d x} [\ln (x)]$

Step 3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{5}{2} x^{\frac{3}{2}} + 2 \frac{1}{x}$

Step 3.3

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

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

Step 4

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

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