prove X^a derived = 1/xlna?

Question

self explanitory please give your answers with working and try and keep it as simple as possible

Answer 1

Pascal is right. If you want to try to involve logs this is what happens...

let y =x^a
Take natural logs of both sides...

lny = alnx

Differentate

(1/y)dy/dx = a/x

ie dy/dx = ay/x

But y=x^a

so dy/dx = ax^a/x = ax^(a-1)
no matter how you do it you get the same answer!!!

Answer 2

But that's incorrect. d(x^a)/dx = ax^(a-1), not 1/x ln a. Observe:

d(x^a)/dx
d(e^(a ln x))/dx
e^(a ln x) d(a ln x)/dx
x^a d(a ln x)/dx
x^a * a/x
ax^(a-1)