How do you differentiate # y = (6e^(-7x)+2x)^2# using the chain rule?

1 Answer
Sep 15, 2016

Answer:

#y'=-504e^(-14x)+12e^(-7x)-84xe^(-7x)+4x#

Explanation:

To differentiate the given function #y# using chain rule let:
#f(x)=x^2# and
#g(x)=6e^(-7x)+2x#
So, #y=f(g(x))#

To differentiate #y=f(g(x))# we have to use chain rule as follows:
Then #y'=(f(g(x)))'=f'(g(x))*g'(x)#

Let's find #f'(x)# and #g'(x)#
#f'(x)=2x#
#g'(x)=-7*6e^(-7x)+2=-42e^(-7x)+2#

#y'=(f(g(x)))'=f'(g(x))*g'(x)#

#y'=2(6e^(-7x)+2x)*(-42e^(-7x)+2)#
#y'=2(-252e^(-14x)+12e^(-7x)-84xe^(-7x)+4x)#
#y'=-504e^(-14x)+12e^(-7x)-84xe^(-7x)+4x#