Calculus Examples

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

Step 1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3} {(x - 1)}^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [{(x - 1)}^{\frac{3}{2}}]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = x - 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $3$ .

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

Step 7

Simplify terms.

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Step 7.1

Combine $\frac{3}{2}$ and ${(x - 1)}^{\frac{1}{2}}$ .

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

Step 7.2

Multiply $\frac{3 {(x - 1)}^{\frac{1}{2}}}{2}$ by $\frac{2}{3}$ .

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

Step 7.3

Multiply.

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Step 7.3.1

Multiply $2$ by $3$ .

$\frac{6 {(x - 1)}^{\frac{1}{2}}}{2 \cdot 3} \frac{d}{d x} [x - 1]$

Step 7.3.2

Multiply $2$ by $3$ .

$\frac{6 {(x - 1)}^{\frac{1}{2}}}{6} \frac{d}{d x} [x - 1]$

Step 7.4

Cancel the common factor.

$\frac{6 {(x - 1)}^{\frac{1}{2}}}{6} \frac{d}{d x} [x - 1]$

Step 7.5

Divide ${(x - 1)}^{\frac{1}{2}}$ by $1$ .

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

Step 8

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 9

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

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

Step 10

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

${(x - 1)}^{\frac{1}{2}} (1 + 0)$

Step 11

Simplify the expression.

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Step 11.1

Add $1$ and $0$ .

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

Step 11.2

Multiply ${(x - 1)}^{\frac{1}{2}}$ by $1$ .

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