Question

Hi, could someone help me differentiate Q6 b with the use if ln​

Answer 1

`\displaystyle (dy)/(dx) = (-(2x - 3)(6x - 43))/((3x + 4)^4)`

General Formulas and Concepts:

Pre-Algebra

• Equality Properties

Algebra II

• Natural logarithms ln and Euler's number e
• Logarithmic Property [Dividing]:
  `\displaystyle log((a)/(b)) = log(a) - log(b)`
• Logarithmic Property [Exponential]:
  `\displaystyle log(a^b) = b \cdot log(a)`

Calculus

Differentiation

• Derivatives
• Derivative Notation
• Implicit Differentiation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. 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))`

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = ((2x - 3)^2)/((3x + 4)^3)`

Step 2: Rewrite

1. [Equality Property] ln both sides:
   `\displaystyle lny = ln \bigg[ ((2x - 3)^2)/((3x + 4)^3) \bigg]`
2. Expand [Logarithmic Property - Dividing]:
   `\displaystyle lny = ln(2x - 3)^2 - ln(3x + 4)^3`
3. Simplify [Logarithmic Property - Exponential]:
   `\displaystyle lny = 2ln(2x - 3) - 3ln(3x + 4)`

Step 3: Differentiate

1. Implicit Differentiation:
   `\displaystyle (dy)/(dx)[lny] = (dy)/(dx) \bigg[ 2ln(2x - 3) - 3ln(3x + 4) \bigg]`
2. 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]`
3. Basic Power Rule:
   `\displaystyle (1)/(y) \ (dy)/(dx) = 4 \bigg( (1)/(2x - 3) \bigg) - 9 \bigg( (1)/(3x + 4) \bigg)`
4. Simplify:
   `\displaystyle (1)/(y) \ (dy)/(dx) = (4)/(2x - 3) - (9)/(3x + 4)`
5. Isolate
   `\displaystyle (dy)/(dx)`:
   `\displaystyle (dy)/(dx) = y \bigg( (4)/(2x - 3) - (9)/(3x + 4) \bigg)`
6. Substitute in y [Derivative]:
   `\displaystyle (dy)/(dx) = ((2x - 3)^2)/((3x + 4)^3) \bigg( (4)/(2x - 3) - (9)/(3x + 4) \bigg)`
7. Simplify:
   `\displaystyle (dy)/(dx) = ((2x - 3)^2)/((3x + 4)^3) \bigg[ (4(3x + 4) - 9(2x - 3))/((2x - 3)(3x +4)) \bigg]`
8. Simplify:
   `\displaystyle (dy)/(dx) = (-(2x - 3)(6x - 43))/((3x + 4)^4)`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Book: College Calculus 10e