How do you find the derivative of #y= ((e^x)/(x^2)) #?

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Answer 1 · 2016-01-13T10:13:51

#(e^x (x-2) ) / (x^3)# with the quotient rule.

You can use the quotient rule to find the derivative.

If #f(x) = g(x) / (h(x))#, then the derivative is:

#f'(x) = (h(x) g'(x) - g(x) h'(x) ) / (h^2(x)) #

In your case,

#g(x) = e^x# and #h(x) = x^2#.

The derivatives of #g(x)# and #h(x)# are

#g'(x) = e^x# and #h'(x) = 2x#.

Thus, according to the formula, you derivative is:

#f'(x) = (x^2 * e^x - e^x * 2x ) / (x^2)^2 = (x e^x (x - 2)) / (x^4)#

... cancel #x# both in the numerator and the denominator...

#= (cancel(color(blue)(x)) e^x (x - 2)) / (x^3 * cancel(color(blue)(x))) = (e^x (x-2) ) / (x^3) #