Answer:
[tex]\displaystyle \frac{dy}{dx} = \frac{-(2x - 3)(6x - 43)}{(3x + 4)^4}[/tex]
General Formulas and Concepts:
Pre-Algebra
Algebra II
- Natural logarithms ln and Euler's number e
- Logarithmic Property [Dividing]: [tex]\displaystyle log(\frac{a}{b}) = log(a) - log(b)[/tex]
- Logarithmic Property [Exponential]: [tex]\displaystyle log(a^b) = b \cdot log(a)[/tex]
Calculus
Differentiation
- Derivatives
- Derivative Notation
- Implicit Differentiation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = \frac{(2x - 3)^2}{(3x + 4)^3}[/tex]
Step 2: Rewrite
- [Equality Property] ln both sides: [tex]\displaystyle lny = ln \bigg[ \frac{(2x - 3)^2}{(3x + 4)^3} \bigg][/tex]
- Expand [Logarithmic Property - Dividing]: [tex]\displaystyle lny = ln(2x - 3)^2 - ln(3x + 4)^3[/tex]
- Simplify [Logarithmic Property - Exponential]: [tex]\displaystyle lny = 2ln(2x - 3) - 3ln(3x + 4)[/tex]
Step 3: Differentiate
- Implicit Differentiation: [tex]\displaystyle \frac{dy}{dx}[lny] = \frac{dy}{dx} \bigg[ 2ln(2x - 3) - 3ln(3x + 4) \bigg][/tex]
- Logarithmic Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle \frac{1}{y} \ \frac{dy}{dx} = 2 \bigg( \frac{1}{2x - 3} \bigg)\frac{dy}{dx}[2x - 3] - 3 \bigg( \frac{1}{3x + 4} \bigg) \frac{dy}{dx}[3x + 4][/tex]
- Basic Power Rule: [tex]\displaystyle \frac{1}{y} \ \frac{dy}{dx} = 4 \bigg( \frac{1}{2x - 3} \bigg) - 9 \bigg( \frac{1}{3x + 4} \bigg)[/tex]
- Simplify: [tex]\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{4}{2x - 3} - \frac{9}{3x + 4}[/tex]
- Isolate [tex]\displaystyle \frac{dy}{dx}[/tex]: [tex]\displaystyle \frac{dy}{dx} = y \bigg( \frac{4}{2x - 3} - \frac{9}{3x + 4} \bigg)[/tex]
- Substitute in y [Derivative]: [tex]\displaystyle \frac{dy}{dx} = \frac{(2x - 3)^2}{(3x + 4)^3} \bigg( \frac{4}{2x - 3} - \frac{9}{3x + 4} \bigg)[/tex]
- Simplify: [tex]\displaystyle \frac{dy}{dx} = \frac{(2x - 3)^2}{(3x + 4)^3} \bigg[ \frac{4(3x + 4) - 9(2x - 3)}{(2x - 3)(3x +4)} \bigg][/tex]
- Simplify: [tex]\displaystyle \frac{dy}{dx} = \frac{-(2x - 3)(6x - 43)}{(3x + 4)^4}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e
