Find the derivative of the inverse function at the indicated point

Question

If f(x) = x^3 + 7x + 2 , find (f^-1)’(1)

Answer 1

f(x) = x³ + 7x + 2

∴ f(-47/330) = 1

∴ f^-1(1) = -47/330 ——————— (1)

f(x) = x³ + 7x + 2

∴ f'(x) = 3x² + 7

∴ f'(-47/330) = 256309/36300 ——————— (2)

Let, f^-1(x) = g(x)

We know, f(f^-1(x)) = x

∴ f(g(x)) = x

d[f(g(x))]

⇒ ----------- = 1

dx

⇒ f'(g(x)) · g'(x) = 1

Putting x = 1, we get

f'(g(1)) · g'(1) = 1

f'(f^-1(1)) · (f^-1)'(1) = 1

f'(-47/330) · (f^-1)'(1) = 1 [from equation (1)]

256309/36300 · (f^-1)'(1) = 1 [from equation (2)]

(f^-1)'(1) = 36300/256309