If f(x) = x^3+x , and a = 10 find (f^-1)'(a) = ?

Answer 1

Step 1: first, find x so that f(x) = x³ + x = 10.

By inspection, we see that f(2) = 10

Step 2: Find f'(x) and evaluate when x = 2

f'(x) = 3x² + 1. So, f'(2) = 13

Step 3: (f^-1)'(10) = 1/f'(2) = 1/13

In other words, the slope of the tangent to the curve y = f^-1(x) at the point (10, 2) is the reciprocal of the slope of the tangent to the curve y = f(x) at the point (2,10).