Calculus Examples

$y = \ln ({(3 - x)}^{\frac{3}{2}})$

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 $g (x) = {(3 - x)}^{\frac{3}{2}}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(3 - x)}^{\frac{3}{2}}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with ${(3 - x)}^{\frac{3}{2}}$ .

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

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 $g (x) = 3 - x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $3 - x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $3 - x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $3$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify the denominator.

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.1.3

Combine the numerators over the common denominator.

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

Step 10.1.4

Add $- 1$ and $3$ .

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

Step 10.1.5

Divide $2$ by $2$ .

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

Step 10.2

Simplify $2 {(3 - x)}^{1}$ .

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

Step 11

By the Sum Rule, the derivative of $3 - x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x]$ .

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

Step 12

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{3}{2 (3 - x)} (0 + \frac{d}{d x} [- x])$

Step 13

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Combine fractions.

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Step 16.1

Multiply $- 1$ by $1$ .

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

Step 16.2

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

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

Step 16.3

Simplify the expression.

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Step 16.3.1

Multiply $3$ by $- 1$ .

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

Step 16.3.2

Move the negative in front of the fraction.

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