Note that f(x) is a compound function envolving the functions g(x) = x^2 and h(x) = 1 - e^(x) and p(x) = 3sqrt(x). In this sense, we could write that f(x) = g(h(p(x))).
We want to find the derivative (df)/(dx) of the function f(x) = g(h(x)). Using the chain rule:
f'(x) = g'[(h(p(x)))] * h'[(p(x))] * p'(x).
Let us calculate each term separately.
1) g'[(h(p(x)))] = 2(1 - e^(3sqrt(x)));
2) h'[(p(x))] = -e^(3sqrt(x));
3) p'(x) = 3/2x^(-1/2).
Then:
f'(x) = cancel(2)(1 - e^(3sqrt(x))) * (-e^(3sqrt(x))) * (3/cancel(2)x^(-1/2));
f'(x) = 3/sqrt(x)e^(3sqrt(x))(e^(3sqrt(x)) - 1);
f'(x) = (3e^(6sqrt(x)))/sqrt(x)(1 - e^(-3sqrt(x))).
Hope it helped you!
