4.) differentiate this transcendental function

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)))