×

Hello! Socratic's Terms of Service and Privacy Policy have been updated, which will be automatically effective on October 6, 2018. Please contact hello@socratic.com with any questions.

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

2 Answers
Jun 17, 2018

Answer:

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

Explanation:

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]

Jun 17, 2018

Answer:

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

Explanation:

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)#