Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{\frac{5}{2}}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $x^{\frac{5}{2}}$ .

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

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{\frac{5}{2}}]$

Step 1.3

Replace all occurrences of $u$ with $x^{\frac{5}{2}}$ .

$\frac{1}{x^{\frac{5}{2}}} \frac{d}{d x} [x^{\frac{5}{2}}]$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $5$ .

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

Step 7

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

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

Step 8

Multiply $\frac{1}{x^{\frac{5}{2}}}$ by $\frac{5 x^{\frac{3}{2}}}{2}$ .

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

Step 9

Simplify the expression.

Tap for more steps...

Step 9.1

Move $2$ to the left of $x^{\frac{5}{2}}$ .

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

Step 9.2

Move $x^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 10

Simplify the denominator.

Tap for more steps...

Step 10.1

Multiply $x^{\frac{5}{2}}$ by $x^{- \frac{3}{2}}$ by adding the exponents.

Tap for more steps...

Step 10.1.1

Move $x^{- \frac{3}{2}}$ .

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

Step 10.1.2

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

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

Step 10.1.3

Combine the numerators over the common denominator.

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

Step 10.1.4

Add $- 3$ and $5$ .

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

Step 10.1.5

Divide $2$ by $2$ .

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

Step 10.2

Simplify $2 x^{1}$ .

$\frac{5}{2 x}$