Answer:

General Formulas and Concepts:
Pre-Algebra
Algebra II
- Natural logarithms ln and Euler's number e
- Logarithmic Property [Dividing]:

- Logarithmic Property [Exponential]:

Calculus
Differentiation
- Derivatives
- Derivative Notation
- Implicit Differentiation
Derivative Property [Multiplied Constant]:
![\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)](https://img.qammunity.org/2022/formulas/mathematics/high-school/rwpyhrof52dro5d128gleq5obchnuu5qkj.png)
Derivative Property [Addition/Subtraction]:
![\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]](https://img.qammunity.org/2022/formulas/mathematics/high-school/i90hl6t3gcguvrecodn8t9gnodav0w5ns8.png)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Quotient Rule]:
![\displaystyle (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))](https://img.qammunity.org/2022/formulas/mathematics/high-school/hrfl3gpx3dh352g7a9uj6guyxz9uxwhvl3.png)
Derivative Rule [Chain Rule]:
![\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)](https://img.qammunity.org/2022/formulas/mathematics/high-school/vue68srn3fe6bds4idxorm97z7tgwelamw.png)
Explanation:
Step 1: Define
Identify

Step 2: Rewrite
- [Equality Property] ln both sides:
![\displaystyle lny = ln \bigg[ ((2x - 3)^2)/((3x + 4)^3) \bigg]](https://img.qammunity.org/2022/formulas/mathematics/college/uz74owawui9efecwzmja7lyt9n3nhrn93k.png)
- Expand [Logarithmic Property - Dividing]:

- Simplify [Logarithmic Property - Exponential]:

Step 3: Differentiate
- Implicit Differentiation:
![\displaystyle (dy)/(dx)[lny] = (dy)/(dx) \bigg[ 2ln(2x - 3) - 3ln(3x + 4) \bigg]](https://img.qammunity.org/2022/formulas/mathematics/college/5xf0lr7mi17qoj6dtcgcld45x6lhy3ozkh.png)
- Logarithmic Differentiation [Derivative Rule - Chain Rule]:
![\displaystyle (1)/(y) \ (dy)/(dx) = 2 \bigg( (1)/(2x - 3) \bigg)(dy)/(dx)[2x - 3] - 3 \bigg( (1)/(3x + 4) \bigg) (dy)/(dx)[3x + 4]](https://img.qammunity.org/2022/formulas/mathematics/college/h8gy128w9d4f72r1c31afb378w57cme0ql.png)
- Basic Power Rule:

- Simplify:

- Isolate
:

- Substitute in y [Derivative]:

- Simplify:
![\displaystyle (dy)/(dx) = ((2x - 3)^2)/((3x + 4)^3) \bigg[ (4(3x + 4) - 9(2x - 3))/((2x - 3)(3x +4)) \bigg]](https://img.qammunity.org/2022/formulas/mathematics/college/8znw9djibsfn46ekvib33k2jbq9gxbl0wd.png)
- Simplify:

Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e