What is the derivative of #f(x)=(e^(2x))(ln(x))#?

Answer 1

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

The derivative of #lnx# is #1/x# The derivative of #e^g(x)#is #e^g(x)*g'(x)# The derivative of #h(x)*l(x)# is #h'(x)*l(x)+h(x)*l'(x)#

Then

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

#=e^(2x)(2lnx+1/x)#

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Answer 2

Evelyn Agard

To find the derivative of ( f(x) = e^{2x} \ln(x) ), you can use the product rule. The product rule states that if you have two functions, ( u(x) ) and ( v(x) ), then the derivative of their product is ( u'(x)v(x) + u(x)v'(x) ). Applying the product rule to the given function, we get:

[ f'(x) = \frac{d}{dx}\left(e^{2x}\right)\ln(x) + e^{2x}\frac{d}{dx}\left(\ln(x)\right) ]

Using the chain rule, ( \frac{d}{dx}(e^{2x}) = 2e^{2x} ), and using the derivative of ( \ln(x) ), which is ( \frac{1}{x} ), we get:

[ f'(x) = 2e^{2x}\ln(x) + e^{2x} \frac{1}{x} ]

So, the derivative of ( f(x) = e^{2x} \ln(x) ) is ( f'(x) = 2e^{2x}\ln(x) + \frac{e^{2x}}{x} ).

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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.