How do you derive #y = x^3+(3x)/e^(x^2)# using the quotient rule?

1 Answer
Bio
Oct 31, 2015

#frac{dy}{dx}=3(x^2+\frac{1-2x^2}{e^{x^2}})#

Explanation:

#frac{dy}{dx}=frac{d}{dx}(x^3+frac{3x}{e^{x^2}})#
#=frac{d}{dx}(x^3)+frac{d}{dx}(frac{3x}{e^{x^2}})#
#=3x^2+frac{d}{dx}(frac{3x}{e^{x^2}})#
#=3x^2+frac{e^{x^2}frac{d}{dx}(3x)-3xfrac{d}{dx}(e^{x^2})}{(e^{x^2})^2}#
#=3x^2+frac{e^{x^2}(3)-3xfrac{d}{dx}(e^{x^2})}{(e^{x^2})^2}#
#=3x^2+frac{e^{x^2}(3)-3x(2xe^{x^2})}{(e^{x^2})^2}#
#=3x^2+frac{e^{x^2}(3)-3x(2xe^{x^2})}{(e^{x^2})^2}frac{e^{-x^2}}{e^{-x^2}}#
#=3x^2+frac{3-3x(2x)}{e^{x^2}}#
#=3x^2+frac{3-6x^2}{e^{x^2}}#
#=3(x^2+\frac{1-2x^2}{e^{x^2}})#

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