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 = e^2x-3 

Answer 1

y = e^2x-3                                     ...(1)
We have to find the inverse function of y = f(x), i.e x in terms of y.
From (1),
2x – 3= log y
∴ 2x = log y + 3
∴ x = f^–1(y)
= `(1)/(2)(log y + 3)`

∴ `"dx"/"dy" = (1)/(2)"d"/"dy"(logy + 3)`

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

= `(1)/(2y)`

= `(1)/(2e^(2x - 3)`                                   ...[By (1)]

∴ `"dy"/"dx" = (1)/(("dx"/"dy")`

= `(1)/(((1)/(2e^(2x - 3)))`
= 2e^2x–3.