How do I use the limit definition of derivative to find #f'(x)# for #f(x)=5x-9x^2# ?

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Answer 1 · 2014-09-19T23:06:36

Let us find #f'(x)# by using the limit definition.

Start with #f(x)#.

#f(x)=5x-9x^2#

Let us find #f(x+h)#.

#f(x+h)=5(x+h)-9(x+h)^2#
#=5x+5h-9(x^2+2xh+h^2)#
#=5x+5h-9x^2-18xh-9h^2#

Let us find the difference quotient.

#{f(x+h)-f(x)}/h#

by plugging in the expression we found above,

#={5x+5h-9x^2-18xh-9h^2-(5x-9x^2)}/h#

by cancelling out #5x#'s and #-9x^2#'s,

#={5h-18xh-9h^2}/h#

by factoring #h# out in the numerator,

#={h(5-18x-9h)}/h#

by cancelling out #h#'s,

#=5-18x-9h#

Now, we can find #f'(x)#.

#f'(x)=lim_{h to 0}(5-18x-9h)=5-18x-9(0)=5-18x#