Question

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

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} = \frac{-(2x - 3)(6x - 43)}{(3x + 4)^4}$

General Formulas and Concepts:

Pre-Algebra

• Equality Properties

Algebra II

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

Calculus

Differentiation

• Derivatives
• Derivative Notation
• Implicit Differentiation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:                                                                           $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Derivative Rule [Chain Rule]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = \frac{(2x - 3)^2}{(3x + 4)^3}$

Step 2: Rewrite

1. [Equality Property] ln both sides:                                                                 $\displaystyle lny = ln \bigg[ \frac{(2x - 3)^2}{(3x + 4)^3} \bigg]$
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:                                                                                 $\displaystyle \frac{dy}{dx}[lny] = \frac{dy}{dx} \bigg[ 2ln(2x - 3) - 3ln(3x + 4) \bigg]$
2. Logarithmic Differentiation [Derivative Rule - Chain Rule]:                       $\displaystyle \frac{1}{y} \ \frac{dy}{dx} = 2 \bigg( \frac{1}{2x - 3} \bigg)\frac{dy}{dx}[2x - 3] - 3 \bigg( \frac{1}{3x + 4} \bigg) \frac{dy}{dx}[3x + 4]$
3. Basic Power Rule:                                                                                         $\displaystyle \frac{1}{y} \ \frac{dy}{dx} = 4 \bigg( \frac{1}{2x - 3} \bigg) - 9 \bigg( \frac{1}{3x + 4} \bigg)$
4. Simplify:                                                                                                         $\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{4}{2x - 3} - \frac{9}{3x + 4}$
5. Isolate  $\displaystyle \frac{dy}{dx}$:                                                                                                     $\displaystyle \frac{dy}{dx} = y \bigg( \frac{4}{2x - 3} - \frac{9}{3x + 4} \bigg)$
6. Substitute in y [Derivative]:                                                                           $\displaystyle \frac{dy}{dx} = \frac{(2x - 3)^2}{(3x + 4)^3} \bigg( \frac{4}{2x - 3} - \frac{9}{3x + 4} \bigg)$
7. Simplify:                                                                                                         $\displaystyle \frac{dy}{dx} = \frac{(2x - 3)^2}{(3x + 4)^3} \bigg[ \frac{4(3x + 4) - 9(2x - 3)}{(2x - 3)(3x +4)} \bigg]$
8. Simplify:                                                                                                         $\displaystyle \frac{dy}{dx} = \frac{-(2x - 3)(6x - 43)}{(3x + 4)^4}$

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

Unit: Differentiation

Book: College Calculus 10e