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_2(x/2)`

Answer 1

y = `log_2(x/2)`                          ...(1)
We have to find the inverse function of y = f(x), i.e x in terms of y.
From (1),
`x/(2)` = 2^y 
∴ x = 2.2^y = 2^y+1
∴ x = f^–1(y) = 2^y+1 
∴ `"dx"/"dy" = "d"/dy"(2^(y + 1))`

= `2^(y + 1).log2."d"/"dy"(y + 1)`

= `2^(y + 1).log2.(1 + 0)`

= `2^(y + 1).log2`

= `2^(log_2(x/2) + 1).log2`             ...[By (1)]

= `2^(log_2(x/2) + log_2 2).log2`

= `2^(log_2(x/2 xx 2).log2`

= 2^log₂x.log2
= x log 2                        ...[∵ a^log_ax = x]

∴ `"dy"/"dx" = (1)/(("dx"/"dy")`

= `(1)/(xlog2)`.