If #f(x)= csc7 x # and #g(x) = e^(1 +3x ) #, how do you differentiate #f(g(x)) # using the chain rule?

1 Answer
Mar 28, 2017

Answer:

#d/(dx)f(g(x))=-21e^((1+3x))csc(7e^(1+3x))cot(7e^(1+3x))#

Explanation:

As #f(x)=csc7x# and #g(x)=e^(1+3x)#

#f(g(x))=csc(7e^(1+3x))#

The chain in the derivative of #f(g(x))# is in fact longer.

We have #f(x)=csc(u(x))#, #u(g(x))=7g(x)#, #g(x)=e^(v(x))# and #v(x)=1+3x#.

Hence #d/(dx)f(g(x))=(df)/(du)xx(du)/(dg)xx(dg)/(dv)xx(dv)/(dx)#

= #-cscucotuxx7xxe^vxx3#

= #-21e^((1+3x))csc(7e^(1+3x))cot(7e^(1+3x))#