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 $f'(x)$ .

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

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