How do you differentiate #f(x)=x(3x-9)^3# using the chain rule?

1 Answer
Jim G.
Feb 25, 2016

#f'(x) = 3(3x - 9 )^2(4x - 3 )#

Explanation:

The #color(blue) " Product rule " "will have to be used as well as the "#
#color(blue)" chain rule "#

There is a product of 2 functions here , x and #(3x - 9 )^3#

using the #color(blue) " Product rule "#

If f(x) = g(x).h(x) then f'(x) = g(x).h'(x) + h(x).g'(x)

using the#color(blue) " chain rule "#

#d/dx[f(g(x))] = f'(g(x)).g'(x)#

hence: #f'(x) = x d/dx(3x-9)^3 + (3x-9)^3 d/dx(x)#

#=x[3(3x-9)^2 d/dx(3x-9)] + (3x-9)^3 .1 #

#=x[3(3x-9)^2 .3] + (3x-9)^3#

#=9x(3x-9)^2 + (3x-9)^3#

( can be simplified by 'taking out' common factor )

#= (3x-9)^2[9x +3x - 9 ] = (3x-9)^2 . 3(4x-3)#

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