Question

How do you find the derivatives of `y=(3x-2)/(4x+3)` by logarithmic differentiation?

Answer 1

see below

Explanation:

Use the Property `color(red)(log_b(x/y)=log_bx-log_by`

Steps in Logarithmic Differentiation:

1. Take natural logarithms of both sides of an equation y=f(x)   and use the Laws of Logarithms to simplify.
2. Differentiate implicitly with respect to x
3. Solve the resulting equation for y’
   4. Replace y with the original equation

`color(blue)(ln y = ln ((3x-2)/(4x+3))`

`color(blue)(ln y = ln (3x-2)-ln(4x+3))`

`color(blue)(d/dx[ln y = ln (3x-2)-ln(4x+3)]`

`color(blue)(1/y dy/dx = 3/(3x-2)-4/(4x+3)`

`color(blue)(dy/dx = [3/(3x-2)-4/(4x+3)]*y`

`color(blue)(dy/dx = [3/(3x-2)-4/(4x+3)]*[(3x-2)/(4x+3)]`