Step 1: Determine the derivative of #(x - 5)^3# using the chain rule
Let #y =u^3# and #u = x - 5#. The derivative of #y# is #y' = 3u^2# and the derivative of #u# is #1#. Hence, the derivative of #y = (x - 5)^3# is #y' = 3(x - 5)^2 xx 1 =color(red)( 3(x - 5)^2)#
Step 2: Determine the derivative of #f(x)# using the product rule
Let #f(x) = g(x) xx h(x)#. Then our derivative is given by #f'(x) = g'(x) xx h(x) + g(x) xx h'(x)#.
The derivative of #g(x)# is #2x#. The derivative of #h(x)#, as mentioned in step #1# is #3(x - 5)^2#.
Hence, #f'(x) = 2x xx(x - 5)^3 + x^2 xx 3(x - 5)^2 =2x(x - 5)^3 + 3x^2(x - 5)^2 = (x - 5)^2(2x(x - 5) + 3x^2) = (x - 5)^2(2x^2 - 10x + 3x^2) = (x - 5)^2(5x^2 - 10x) =color(blue)( 5x(x - 5)^2(x - 2))#
Hopefully this helps!
