Calculus Log Functions?

Question

Find x for this problem

f(x)=4e^(xsinx)

I thought you might go
ln4e^(xsinx)
with the property you could bring down (xsinx)ln4e which would be (xsinx)ln4+lne but I am not sure where to go from there! Any help would be great!

Answer 1

You're incorrect with your logs anyway:
ln f(x) = ln 4 + (x sin x) ln e = ln 4 + x sin x.
So you can get x sin x = ln f(x) - ln 4 [= ln (f(x) / 4)], but in general there's no way to get an explicit expression for x. You can solve certain particular values, for instance if f(x) = 4 then x sin x = 0 so x = kπ for some integer k. But in general this equation can only be solved numerically.

Answer 2

Do you wish to find f `(x)?
If so:-
Let y = f(x) = 4 e^(x.sin x)
let u = x sin x
du/dx = sin x + x cos x
y = 4 e^(u)
dy/du = 4 e^(u) = 4 e^(x sin x)
dy / dx = (dy/du) x (du / dx)
dy / dx = 4 e^(x sin x) . (sinx + x cos x)