# How do you find the derivative of #y=e^x*lnx#?

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To find the derivative of ( y = e^x \cdot \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, ( u(x) \cdot v(x) ), is given by the formula:

[ (u \cdot v)' = u'v + uv' ]

For ( y = e^x \cdot \ln(x) ), let ( u(x) = e^x ) and ( v(x) = \ln(x) ).

Now, we find the derivatives of ( u(x) ) and ( v(x) ):

[ u'(x) = e^x ] [ v'(x) = \frac{1}{x} ]

Applying the product rule:

[ y' = (e^x \cdot \ln(x))' = u'v + uv' = e^x \cdot \frac{1}{x} + e^x \cdot \ln(x) ]

So, the derivative of ( y = e^x \cdot \ln(x) ) is:

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

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