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\frac{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`