If #f(x)= cos 4 x # and #g(x) = 2 x #, how do you differentiate #f(g(x)) # using the chain rule?

1 Answer
Jun 18, 2016

Answer:

#-8sin(8x)#

Explanation:

The chain rule is stated as:

#color(blue)((f(g(x)))'=f'(g(x))*g'(x))#

Let's find the derivative of #f(x)# and #g(x)#

#f(x)=cos(4x)#
#f(x)=cos(u(x))#

We have to apply chain rule on #f(x)#
Knowing that #(cos(u(x))'=u'(x)*(cos'(u(x))#

Let #u(x)=4x#
#u'(x)=4#

#f'(x)=u'(x)*cos'(u(x))#

#color(blue)(f'(x)=4*(-sin(4x))#

#g(x)=2x#
#color(blue)(g'(x)=2)#

Substituting the values on the property above:

#color(blue)((f(g(x)))'=f'(g(x))*g'(x))#

#(f(g(x)))'=4(-sin(4*(g(x)))*2#
#(f(g(x)))'=4(-sin(4*2x))*2#

#(f(g(x)))'=-8sin(8x)#