Use logarithmic differentiation to find the derivative of the function.

Question

Use logarithmic differentiation to find the derivative of the function.

y = (3x + 2)⁴(2x − 5)²

Answer 1

Logarithmic differentiation is essentially taking the natural log of both sides of the equation, applying a few logarithmic rules, then differentiating and rearranging as necessary.

ln(y) = ln((3x + 2)⁴(2x − 5)²)

You can break up the left side of the equation using the product rule of logarithms:

ln(y) = ln(3x + 2)⁴ + ln((2x − 5)²)

and bring the exponents down.

ln (y) = 4ln(3x+2) + 2(ln(2x - 5))

And now, differentiate both sides; you will need to apply the chain rule.

(1/y)*(dy/dx) = 4*(1/(3x+2))*3 + 2*(1/(2x-5))*2

This expression can be rearranged to solve for the derivative, dy/dx. You can also substitute your original expression for y. This should leave you with

dy/dx = (12/(3x + 2) + 4/(2x - 5)) * (3x + 2)⁴(2x − 5)²