Find derivative? #f(x) = (xe^x)^2#

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Answer 1 · 2018-06-17T12:19:20

# f^'(x)= 2 x e^(2 x)(x + 1) #

Differentiation of product of two functions:

#d/ (d x) (f(x)g(x)) = f^'(x) g(x) + f(x) g'(x)#

#f(x) = (x e^x)^2 or f(x) = x^2 * e^(2 x) #

# f^'(x)= x^2 * e^(2 x)* 2 + 2 x * e^(2 x) #

#:. f^'(x)= 2 x e^(2 x)(x + 1) # [Ans]

Answer 2 · 2018-06-17T12:31:50

#f'(x)=2x(x+1)e^(2x)#

Here,

#f(x)=(xe^x)^2#

Let,

#y=u^2 ,where, color(red)(u=x*e^x)#

#:.(dy)/(du)=2u .......to(1)# .

#" Using "color(blue)"Product Rule" # , we get

#(du)/(dx)=x*d/(dx)(e^x)+e^x*d/(dx)(x)#

#=>(du)/(dx)=x*e^x+e^x*1...to(2)#

Now ,#"using "color(blue)"Chain Rule":#

#color(blue)((dy)/(dx)=(dy)/(du)*(du)/(dx)#

#(dy)/(dx)=(2color(red)(u))xx(xe^x+e^x)....toFrom (1) and (2)#

#:.(dy)/(dx)=2color(red)((xe^x))xx(xe^x+e^x)#

#(dy)/(dx)=2xe^x xx e^x(x+1)#

#(dy)/(dx)=2x(x+1)e^(2x)#
...........................................................................................................

OR

#f(x)=(xe^x)^2=x^2e^(2x)#

#=>f'(x)=x^2d/(dx)(e^(2x))+e^(2x)d/(dx)(x^2)#

#=>f'(x)=x^2e^(2x)*2+e^(2x)(2x)=2x^2e^(2x)+2xe^(2x)#

Hence,

#f'(x)=(2x^2+2x)e^(2x)=2x(x+1)e^(2x)#