Question

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?

Answer 1

`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`