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

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

Lets get started!

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:

Now, sub everything in:

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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To differentiate ( g(x) = e^x \sqrt{1 - e^{2x}} ) using the product rule, let ( u(x) = e^x ) and ( v(x) = \sqrt{1 - e^{2x}} ). Then apply the product rule:

[ g'(x) = u'(x)v(x) + u(x)v'(x) ]

Where ( u'(x) = e^x ) and ( v'(x) = \frac{-e^{2x}}{2\sqrt{1 - e^{2x}}} ).

Plug these into the formula:

[ g'(x) = e^x \sqrt{1 - e^{2x}} + e^x \left( \frac{-e^{2x}}{2\sqrt{1 - e^{2x}}} \right) ]

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