Question

sing the definition of the​ derivative, find f'(x). Then find f'(1)​, f'(2)​, and f'(3)when the derivative exists. f(x)=-x^2+4x-9

Answer 1

By definition of the derivative, and with $f(x)=-x^2+4x-9$, we have

$f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h$

In this case,

$f(x+h)=-(x+h)^2+4(x+h)-9=-x^2-2xh-h^2+4x+4h-9=f(x)+(4-2x)h-h^2$

Then

$f(x+h)-f(x)=(4-2x)h-h^2$

$\implies\dfrac{f(x+h)-f(x)}h=4-2x-h$

so that as $h\to0$, we're left with

$f'(x)=4-2x$

and so

$f'(1)=2$

$f'(2)=0$

$f'(3)=-2$