Calculus Examples

$g (u) = \frac{2 u - 3}{\sqrt{u^{2} - 3 u + 4}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u^{2} - 3 u + 4}$ as ${(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ .

$\frac{d}{d u} [\frac{2 u - 3}{{(u^{2} - 3 u + 4)}^{\frac{1}{2}}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d u} [\frac{f (u)}{g (u)}]$ is $\frac{g (u) \frac{d}{d u} [f (u)] - f (u) \frac{d}{d u} [g (u)]}{{g (u)}^{2}}$ where $f (u) = 2 u - 3$ and $g (u) = {(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ .

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} \frac{d}{d u} [2 u - 3] - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{{({(u^{2} - 3 u + 4)}^{\frac{1}{2}})}^{2}}$

Step 3

Multiply the exponents in ${({(u^{2} - 3 u + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} \frac{d}{d u} [2 u - 3] - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{{(u^{2} - 3 u + 4)}^{\frac{1}{2} \cdot 2}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} \frac{d}{d u} [2 u - 3] - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{{(u^{2} - 3 u + 4)}^{\frac{1}{2} \cdot 2}}$

Step 3.2.2

Rewrite the expression.

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} \frac{d}{d u} [2 u - 3] - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{{(u^{2} - 3 u + 4)}^{1}}$

Step 4

Simplify.

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} \frac{d}{d u} [2 u - 3] - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $2 u - 3$ with respect to $u$ is $\frac{d}{d u} [2 u] + \frac{d}{d u} [- 3]$ .

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} (\frac{d}{d u} [2 u] + \frac{d}{d u} [- 3]) - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 5.2

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

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} (2 \frac{d}{d u} [u] + \frac{d}{d u} [- 3]) - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 5.3

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

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} (2 \cdot 1 + \frac{d}{d u} [- 3]) - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 5.4

Multiply $2$ by $1$ .

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} (2 + \frac{d}{d u} [- 3]) - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 5.5

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

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} (2 + 0) - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $2$ and $0$ .

$\frac{{(u^{2} - 3 u + 4)}^{\frac{1}{2}} \cdot 2 - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 5.6.2

Move $2$ to the left of ${(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ .

$\frac{2 \cdot {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) \frac{d}{d u} [{(u^{2} - 3 u + 4)}^{\frac{1}{2}}]}{u^{2} - 3 u + 4}$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = u^{\frac{1}{2}}$ and $g (u) = u^{2} - 3 u + 4$ .

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Step 6.1

To apply the Chain Rule, set $u_{1}$ as $u^{2} - 3 u + 4$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 6.2

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

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 6.3

Replace all occurrences of $u_{1}$ with $u^{2} - 3 u + 4$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {(u^{2} - 3 u + 4)}^{\frac{1}{2} - 1} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 7

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

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {(u^{2} - 3 u + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 8

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

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {(u^{2} - 3 u + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 9

Combine the numerators over the common denominator.

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {(u^{2} - 3 u + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {(u^{2} - 3 u + 4)}^{\frac{1 - 2}{2}} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {(u^{2} - 3 u + 4)}^{\frac{- 1}{2}} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2} {(u^{2} - 3 u + 4)}^{- \frac{1}{2}} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 11.2

Combine $\frac{1}{2}$ and ${(u^{2} - 3 u + 4)}^{- \frac{1}{2}}$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{{(u^{2} - 3 u + 4)}^{- \frac{1}{2}}}{2} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 11.3

Move ${(u^{2} - 3 u + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} \frac{d}{d u} [u^{2} - 3 u + 4])}{u^{2} - 3 u + 4}$

Step 12

By the Sum Rule, the derivative of $u^{2} - 3 u + 4$ with respect to $u$ is $\frac{d}{d u} [u^{2}] + \frac{d}{d u} [- 3 u] + \frac{d}{d u} [4]$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} (\frac{d}{d u} [u^{2}] + \frac{d}{d u} [- 3 u] + \frac{d}{d u} [4]))}{u^{2} - 3 u + 4}$

Step 13

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

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} (2 u + \frac{d}{d u} [- 3 u] + \frac{d}{d u} [4]))}{u^{2} - 3 u + 4}$

Step 14

Since $- 3$ is constant with respect to $u$ , the derivative of $- 3 u$ with respect to $u$ is $- 3 \frac{d}{d u} [u]$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} (2 u - 3 \frac{d}{d u} [u] + \frac{d}{d u} [4]))}{u^{2} - 3 u + 4}$

Step 15

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

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} (2 u - 3 \cdot 1 + \frac{d}{d u} [4]))}{u^{2} - 3 u + 4}$

Step 16

Multiply $- 3$ by $1$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} (2 u - 3 + \frac{d}{d u} [4]))}{u^{2} - 3 u + 4}$

Step 17

Since $4$ is constant with respect to $u$ , the derivative of $4$ with respect to $u$ is $0$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} (2 u - 3 + 0))}{u^{2} - 3 u + 4}$

Step 18

Add $2 u - 3$ and $0$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - (2 u - 3) (\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} (2 u - 3))}{u^{2} - 3 u + 4}$

Step 19

Raise $2 u - 3$ to the power of $1$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - ({(2 u - 3)}^{1} (2 u - 3)) \frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 20

Raise $2 u - 3$ to the power of $1$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - ({(2 u - 3)}^{1} {(2 u - 3)}^{1}) \frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 21

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

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - {(2 u - 3)}^{1 + 1} \frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 22

Add $1$ and $1$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - {(2 u - 3)}^{2} \frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 23

Combine $\frac{1}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}$ and ${(2 u - 3)}^{2}$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - \frac{{(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 24

To write $2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}$ .

$\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} \cdot \frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} - \frac{{(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 25

Combine $2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ and $\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}$ .

$\frac{\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} (2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}})}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} - \frac{{(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 26

Combine the numerators over the common denominator.

$\frac{\frac{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} (2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 27

Multiply $2$ by $2$ .

$\frac{\frac{4 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} {(u^{2} - 3 u + 4)}^{\frac{1}{2}} - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 28

Multiply ${(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ by ${(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 28.1

Move ${(u^{2} - 3 u + 4)}^{\frac{1}{2}}$ .

$\frac{\frac{4 ({(u^{2} - 3 u + 4)}^{\frac{1}{2}} {(u^{2} - 3 u + 4)}^{\frac{1}{2}}) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 28.2

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

$\frac{\frac{4 {(u^{2} - 3 u + 4)}^{\frac{1}{2} + \frac{1}{2}} - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 28.3

Combine the numerators over the common denominator.

$\frac{\frac{4 {(u^{2} - 3 u + 4)}^{\frac{1 + 1}{2}} - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 28.4

Add $1$ and $1$ .

$\frac{\frac{4 {(u^{2} - 3 u + 4)}^{\frac{2}{2}} - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 28.5

Divide $2$ by $2$ .

$\frac{\frac{4 {(u^{2} - 3 u + 4)}^{1} - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 29

Simplify $4 {(u^{2} - 3 u + 4)}^{1}$ .

$\frac{\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$

Step 30

Rewrite $\frac{\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}}{u^{2} - 3 u + 4}$ as a product.

$\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}} \cdot \frac{1}{u^{2} - 3 u + 4}$

Step 31

Multiply $\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}}}$ by $\frac{1}{u^{2} - 3 u + 4}$ .

$\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{1}{2}} (u^{2} - 3 u + 4)}$

Step 32

Raise $u^{2} - 3 u + 4$ to the power of $1$ .

$\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 ({(u^{2} - 3 u + 4)}^{1} {(u^{2} - 3 u + 4)}^{\frac{1}{2}})}$

Step 33

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

$\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{1 + \frac{1}{2}}}$

Step 34

Write $1$ as a fraction with a common denominator.

$\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 35

Combine the numerators over the common denominator.

$\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{2 + 1}{2}}}$

Step 36

Add $2$ and $1$ .

$\frac{4 (u^{2} - 3 u + 4) - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37

Simplify.

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Step 37.1

Apply the distributive property.

$\frac{4 u^{2} + 4 (- 3 u) + 4 \cdot 4 - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2

Simplify the numerator.

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Step 37.2.1

Simplify each term.

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Step 37.2.1.1

Multiply $- 3$ by $4$ .

$\frac{4 u^{2} - 12 u + 4 \cdot 4 - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.2

Multiply $4$ by $4$ .

$\frac{4 u^{2} - 12 u + 16 - {(2 u - 3)}^{2}}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.3

Rewrite ${(2 u - 3)}^{2}$ as $(2 u - 3) (2 u - 3)$ .

$\frac{4 u^{2} - 12 u + 16 - ((2 u - 3) (2 u - 3))}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.4

Expand $(2 u - 3) (2 u - 3)$ using the FOIL Method.

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Step 37.2.1.4.1

Apply the distributive property.

$\frac{4 u^{2} - 12 u + 16 - (2 u (2 u - 3) - 3 (2 u - 3))}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.4.2

Apply the distributive property.

$\frac{4 u^{2} - 12 u + 16 - (2 u (2 u) + 2 u \cdot - 3 - 3 (2 u - 3))}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.4.3

Apply the distributive property.

$\frac{4 u^{2} - 12 u + 16 - (2 u (2 u) + 2 u \cdot - 3 - 3 (2 u) - 3 \cdot - 3)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5

Simplify and combine like terms.

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Step 37.2.1.5.1

Simplify each term.

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Step 37.2.1.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 u^{2} - 12 u + 16 - (2 \cdot 2 u \cdot u + 2 u \cdot - 3 - 3 (2 u) - 3 \cdot - 3)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5.1.2

Multiply $u$ by $u$ by adding the exponents.

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Step 37.2.1.5.1.2.1

Move $u$ .

$\frac{4 u^{2} - 12 u + 16 - (2 \cdot 2 (u \cdot u) + 2 u \cdot - 3 - 3 (2 u) - 3 \cdot - 3)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5.1.2.2

Multiply $u$ by $u$ .

$\frac{4 u^{2} - 12 u + 16 - (2 \cdot 2 u^{2} + 2 u \cdot - 3 - 3 (2 u) - 3 \cdot - 3)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5.1.3

Multiply $2$ by $2$ .

$\frac{4 u^{2} - 12 u + 16 - (4 u^{2} + 2 u \cdot - 3 - 3 (2 u) - 3 \cdot - 3)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5.1.4

Multiply $- 3$ by $2$ .

$\frac{4 u^{2} - 12 u + 16 - (4 u^{2} - 6 u - 3 (2 u) - 3 \cdot - 3)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5.1.5

Multiply $2$ by $- 3$ .

$\frac{4 u^{2} - 12 u + 16 - (4 u^{2} - 6 u - 6 u - 3 \cdot - 3)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5.1.6

Multiply $- 3$ by $- 3$ .

$\frac{4 u^{2} - 12 u + 16 - (4 u^{2} - 6 u - 6 u + 9)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.5.2

Subtract $6 u$ from $- 6 u$ .

$\frac{4 u^{2} - 12 u + 16 - (4 u^{2} - 12 u + 9)}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.6

Apply the distributive property.

$\frac{4 u^{2} - 12 u + 16 - (4 u^{2}) - (- 12 u) - 1 \cdot 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.7

Simplify.

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Step 37.2.1.7.1

Multiply $4$ by $- 1$ .

$\frac{4 u^{2} - 12 u + 16 - 4 u^{2} - (- 12 u) - 1 \cdot 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.7.2

Multiply $- 12$ by $- 1$ .

$\frac{4 u^{2} - 12 u + 16 - 4 u^{2} + 12 u - 1 \cdot 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.1.7.3

Multiply $- 1$ by $9$ .

$\frac{4 u^{2} - 12 u + 16 - 4 u^{2} + 12 u - 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.2

Combine the opposite terms in $4 u^{2} - 12 u + 16 - 4 u^{2} + 12 u - 9$ .

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Step 37.2.2.1

Subtract $4 u^{2}$ from $4 u^{2}$ .

$\frac{- 12 u + 16 + 0 + 12 u - 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.2.2

Add $- 12 u + 16$ and $0$ .

$\frac{- 12 u + 16 + 12 u - 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.2.3

Add $- 12 u$ and $12 u$ .

$\frac{0 + 16 - 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.2.4

Add $0$ and $16$ .

$\frac{16 - 9}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$

Step 37.2.3

Subtract $9$ from $16$ .

$\frac{7}{2 {(u^{2} - 3 u + 4)}^{\frac{3}{2}}}$