Answer:
(i)

(ii)

(iii)
![\displaystyle y' = (e^x[xln(2x) + 1])/(x)](https://img.qammunity.org/2022/formulas/mathematics/college/w7m98m3morq0bbh230yg346gudyge8h1lk.png)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
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]:
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]:
![\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)](https://img.qammunity.org/2022/formulas/mathematics/college/c6fshhoq1mws6w0d0la17c7k2dcytwd8kg.png)
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)
Exponential Differentiation
Logarithmic Differentiation
Explanation:
(i)
Step 1: Define
Identify

Step 2: Differentiate
- Product Rule:
![\displaystyle y' = (3x^2 - x)'ln(2x + 1) + (3x^2 - x)[ln(2x + 1)]'](https://img.qammunity.org/2022/formulas/mathematics/college/8ig38iztibi7ofcc2kujagw0zm0ockvqlp.png)
- Basic Power Rule/Logarithmic Differentiation [Chain Rule]:

- Basic Power Rule:

- Simplify [Factor]:

(ii)
Step 1: Define
Identify

Step 2: Differentiate
- Quotient Rule:

- Basic Power Rule/Logarithmic Differentiation:

- Rewrite:

- Simplify:

(iii)
Step 1: Define
Identify

Step 2: Differentiate
- Product Rule:
![\displaystyle y' = (e^x)'ln(2x) + e^x[ln(2x)]'](https://img.qammunity.org/2022/formulas/mathematics/college/5s75wyjrliieakfaa2r4rwxfd7kcu75yjb.png)
- Exponential Differentiation/Logarithmic Differentiation [Chain Rule]:

- Basic Power Rule:

- Simplify:

- Rewrite:

- Factor:
![\displaystyle y' = (e^x[xln(2x) + 1])/(x)](https://img.qammunity.org/2022/formulas/mathematics/college/w7m98m3morq0bbh230yg346gudyge8h1lk.png)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e