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^x – 3

Answer 1

y = e^x – 3 ...(1)

We have to find the inverse function of y = f(x), i.e x in terms of y.

From (1),

e^x = y + 3

∴ x = log(y + 3)

∴ x = f^–1(y) = log(y + 3)

∴ `"dx"/"dy" = "d"/"dy"[log("y" + 3)]`

= `(1)/("y" + 3)."d"/"dy"("y" + 3)`

= `(1)/("y" + 3).(1 + 0)`

= `(1)/("y" + 3)`

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

= `(1)/"e"^"x"`

∴ `"dy"/"dx" = (1)/(("dx"/"dy")`

= `(1)/((1/("e"^"x"))`

= e^x