How do you differentiate #g(x) = e^xsqrt(1-e^(2x))# using the product rule?

1 Answer
Larry Leitch-Casey
Nov 17, 2016

Hey there! To differentiate a function using the product rule, keep note of the general formula for the derivative of a product whereby if:

#f(x) = g(x) * h(x)# then,

#f'(x) = g'(x)*h(x) + h'(x)*g(x)#

Lets get started!

Explanation:

In this example, lets just change #g(x)# to #f(x)# so it fits with the general formula. With this, your g(x) and h(x) are as follows:

#g(x) = e^x#

#h(x) = sqrt(1-e^2x)# which is equivalent to #(1-e^2x)^(1/2)#

Now, if you follow the derivative general formula, it reads "derivative of the 1st, times the 2nd - plus derivative of the 2nd times the 1st. Lets get those derivatives separately:

#g'(x) = e^x -> # Note that the derivative of #e^x# is always #e^x#

#h'(x) = 1/2(1-e^2x)^(-1/2)*(-2e^(2x)) -> # Computed using chain rule!

Now, sub everything in:

#f'(x) = g'(x)*h(x) + h'(x)*g(x)#

#f'(x) = (e^x)((1-e^2x)^(1/2)) +(1/2(1-e^2x)^(-1/2)*(-2e^(2x)))(e^x)#

And that's it! One suggestion I do have; if you can do these "inner" derivative in your head and as you go(i.e. the chain rule we had to do), this will allow you to complete the question much faster. I only did the derivatives separately for demonstrative purposes.

Hopefully this helped and was clear for you! If you have any questions, please let me know! :)

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