The function f(x) = tan(3^x) has one zero in the interval [0, 1.4]. What is the derivative at this point?

The function f(x) = tan(3^x) has one zero in the interval [0, 1.4]. What is the derivative at this point?

1 Answer
Dec 1, 2016

pi ln3

Explanation:

If tan(3^x) = 0, then sin(3^x) = 0 and cos(3^x) = +-1

Therefore 3^x = kpi for some integer k.

We were told that there is one zero on [0,1.4]. That zero is NOT x=0 (since tan 1 != 0). The smallest positive solution must have 3^x = pi.

Hence, x = log_3 pi.

Now let's look at the derivative.

f'(x) = sec^2(3^x) * 3^x ln3

We know from above that 3^x = pi, so at that point

f' = sec^2(pi) * pi ln3 =(-1)^2 pi ln3 = pi ln3