Here is the differentiation rule for exponentials.
d(e^u) = (e^u)·du
We can apply that several times.
Where u = e^v
... du = (e^v)·dv
Where v = e^w
... dv = (e^w)·dw
For w = 4x^2+1
... dw = 8x·dx
Now, we can fill in the links of the chain.
d(e^u) = (e^u)·(e^v)·(e^(4x^2+1))·8x·dx
We can add the exponents of terms with the same base that are multiplied.
... = 8x·e^(u +v +(4x^2+1))·dx
Filling in for u and v, we have
... f'(x) = 8x·e^(1 +4x^2 +e^(1 +4x^2) +e^(e^(1 +4x^2)))
