How do you find the derivative of #f(x)=csc(3x-1)#?

1 Answer
Oct 20, 2016

#f'(x)=3cos(3x-1)csc^2(3x-1)#

Explanation:

#f(x)# is a xomposite of two functions
Let :
#color(blue)(g(x)=3x-1) and color(brown)(h(x)=cscx #
So,
#f(x)=color(brown)hcolor(blue)((g(x))#
so thederivative of this function is by applying chain rule:

#color(red)(f'(x))=color(red)(h'(g(x)*(g'(x))#

#color(red)h'(g(x))=????#

#color(brown)(h(x)=cscx #
#h'(x)=cosx*csc^2x#
#color(red)h'(g(x))=cos(g(x))*csc^2(g(x))=cos(3x-1)*csc^2(3x-1)#

#color(red)(h'(g(x))) = cos(3x-1)*csc^2(3x-1)#

#color(red)(g'(x))=????#

#color(blue)(g(x)=3x-1)#
#color(red)(g'(x))=3#

#color(red)(f'(x))=color(red)(h'(g(x)*(g'(x))#
#color(red)(f'(x))=cos(3x-1)*csc^2(3x-1)*3#

#color(red)(f'(x)=3cos(3x-1)csc^2(3x-1))#