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Answer 1 · 2017-09-27T03:16:14

#dy/dx = (-2e^(2/x))/x^2#

Assuming you are looking for #d/dx (e^(2/x))#, typically this is done with a logarithm and implicit differentiation by many people.

Begin by writing the function in terms of #x# and #y#:

#y = e^(2/x)#

Now, take the natural log of both sides:

#ln(y) = ln(e^(2/x))#
#ln(y) = (2/x) * ln(e)#
#ln(y) = 2/x#

Differentiate implicitly, and solve for #dy/dx#:

#d/dx(ln(y)) = d/dx(2/x)#
#(1/y)dy/dx = -2/x^2#

#dy/dx = (-2/x^2)*y = (-2/x^2)*e^(2/x)=(-2e^(2/x))/x^2#