How do you differentiate # f(x)=(1-e^(3sqrtx))^2# using the chain rule.?

Answer 1

#f'(x) = (3e^(6sqrt(x)))/sqrt(x)(1 - e^(-3sqrt(x))).#

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!

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.