Question

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f^–1(y) in the following:

y = log(2x – 1)

Answer 1

y = log(2x – 1)                                     ...(1)
We have to find the inverse function of y = f(x), i.e x in terms of y.
From (1),
2x – 1 = e^y
∴ 2x = e^y + 1
∴ x = f^–1(y)
= `(1)/(2)(e^y + 1)`

∴ `"dx"/"dy" = (1)/(2)"d"/"dy"(e^y + 1)`

= `(1)/(2)(e^y + 0)`

= `(1)/(2)e^y`

= `(1)/(2)e^(log(2x - 1)`                      ...[By (1)]

= `(1)/(2)(2x - 1)`                       ...[∵ e^logx = x]

∴ `"dy"/"dx" = (1)/(("dx"/"dy")`

= `(2)/(2x - 1)`