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!
