Find the derivative of g(x)=x2log3√(x-1).

Question

Find the derivative of g(x)=x2log3√(x-1). Please show all steps.

Answer 1

I was preparing an answer when I found John's answer, which unfortunately has an error. d/dx log_a x = 1/x •1/ln a, so the 2nd term of the product rule derivative should be (x² /2) [1/(x-1)] (1/ln 3) = x² /[2 (x-1) ln 3], that is, the x-1 belongs in the denominator. It is clever how John simplified the combination 2 ln 3 in the denominator into ln 9! So the full answer is g'(x) = log₃(x-1)^x + x² /[(x-1) ln 9].

Answer 2

g(x)=x2log3√(x-1)

g(x) = simplify: x²log₃(x-1)^1/2 = x²(1/2)log₃(x-1) = (x²/2) log₃(x-1)

g'(x) = (2x/2) log₃(x-1) + (x²/2)(x-1)/ln3

= (x) log₃(x-1) + x²(x-1)/(2ln3)

= log₃(x-1)^x + x²(x-1)/ln9