If #f(x)= - e^(5x # and #g(x) = 2x^3 #, how do you differentiate #f(g(x)) # using the chain rule?

1 Answer
Dec 4, 2016

Answer:

#(f(g(x)))'= -30x^2e^(10x^3)#

Explanation:

#f(g(x))# is a composite function, so differentiating it is
#" "#
determined by applying chain rule.
#" "#
Chain rule :
#" "#
#(f(g(x)))'= f'(g(x))xxg'(x)#
#" "#
#" "#
Let us compute #" "color(blue)(f'(g((x)))#
#" "#
Knowing the differentiation of #e^u#:
#" "#
#(e^u)' = u'e^u#
#" "#
#f'(x)=-5e^(5x)" " #
#" "#
then
#" "#
#f'(g(x)))=-5e^(5(g(x)))#
#" "#
#color(blue)(f'(g(x)))=-5e^(5(2x^3))#
#" "#
#color(blue)(f'(g(x))=-5e^(10x^3))#
#" "#
#" "#
Differentiation of #g(x)#
#" "#
#g'(x)# is determined by applying the power rule.
#" "#
Power Rule:
#" "#
#color(red)((x^n)' = nx^(n-1))#
#" "#
#g'(x)=2(x^3)'=2(color(red)(3x^2))=6x^2#
#" "#
Therefore ,
#" "#
#(f(g(x)))'= f'(g(x))xxg'(x)#
#" "#
#(f(g(x)))'= -5e^(10x^3)xx6x^2#
#" "#
#(f(g(x)))'= -30x^2e^(10x^3)#