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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 = ex – 3
Solution
y = ex – 3 ...(1)
We have to find the inverse function of y = f(x), i.e x in terms of y.
From (1),
ex = 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"))`
= ex
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