Question

Let g(x) be the inverse of an invertible function f(x) which is derivable at x = 3. If f(3) = 9 and f’(3) = 9, write the value of g’(9).

Answer 1

From the definition of invertible function,

g(f(x)) = x …(i)

So, g(f(3)) = 3, i.e., g(9) = 3

Now, differentiating both sides of equation (i) w.r.t. x using the Chain Rule of Differentiation, we get –

g’(f(x)). f’(x) = 1 …(ii)

Plugging in x = 3 in equation (ii) gives us –

g’(f(3)).f’(3) = 1

or, g’(9).9 = 1

i.e., g’(9) = 1/9