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

1 Answer

Answer:

#f' (g(x))=-3/sqrt(2x-3)*csc (3sqrt(2x-3) )*cot (3sqrt(2x-3))#

Explanation:

Given #f(x)=csc 3x# and #g(x) = sqrt(2x-3)#

#f(g(x))= csc 3g(x)#
#f(g(x))=csc 3sqrt(2x-3)#

The formula for derivative of #csc u:#

#d/dx(csc u) = - csc u*cot u* (du)/dx#

Let #u=3sqrt(2x-3)#

Take note:
#d/dx(3 sqrt(2x-3))=3* 1/(2sqrt(2x-3))*2#

#f '(g(x))=#
#- csc (3 sqrt(2x-3)) * cot (3 sqrt(2x-3)) * 3* 1/(2sqrt(2x-3))*2#

#f' (g(x))=-3/sqrt(2x-3)*csc (3sqrt(2x-3) )*cot (3sqrt(2x-3))#