How do you differentiate #f(x)=e^x*e^(1/x) # using the product rule?
1 Answer
Mar 6, 2016
Explanation:
The Product Rule:
#frac{"d"}{"d"x}(uv) = v frac{"d"u}{"d"x} + u frac{"d"v}{"d"x}#
In this question,
#u = e^x#
#frac{"d"u}{"d"x} = e^x#
#v = e^{1/x}#
#frac{"d"v}{"d"x} = e^{1/x} frac{"d"}{"d"x}(1/x)#
#= -1/x^2 e^{1/x}#
So,
#f'(x) = v frac{"d"u}{"d"x} + u frac{"d"v}{"d"x}#
#= e^{1/x} * e^x + e^x * (-1/x^2 e^{1/x})#
#= e^{1/x} * e^x * (1 - 1/x^2)#
