If #f(x)= - x^2 + x # and #g(x) = sqrtx + x #, how do you differentiate #f(g(x)) # using the chain rule?

1 Answer
May 17, 2018

Answer:

#-1-2sqrt(x)-x/sqrt(x)-2x#

Explanation:

Assuming you already know the chain rule. If not:
Given #h(x)# is a composite function consisting of 2 functions such that #h(x)=f(g(x))#, #h'(x)=f'(g(x))g'(x)#.

So, in this scenario, #f'(x)=-2x# and #g'(x)=1/(2(sqrt(x)))+1#
Substitute #g(x)# into #f'(x)#:
#-2(sqrt(x)+x)#
And multiply by #g'(x)#:
#-2(sqrt(x)+x)*(1/(2sqrt(x))+1)#
#-2(1/2+sqrt(x)+x/(2sqrt(x))+x)#
#-1-2sqrt(x)-x/sqrt(x)-2x#