Question

How do you use the definition of the derivative to find f '(x) and f ''(x) for `f(x) = 4 + 9x - x^2`?

Answer 1

See the explanation.

Explanation:

Use the definition of the derivative to find f '(x) and f ''(x) for `f(x) = 4 + 9x - x^2`

Recall the definition of derivative:

`f'(x) = lim_(hrarr0) (f(x+h)-f(x))/h`

So for this function, we have:

`f'(x) = lim_(hrarr0) (overbrace([4+9(x+h)-(x+h)^2])^(f(x+h)) - overbrace([4+9x-x^2])^f(x))/h`.

`= lim_(hrarr0) ([4+9x+9h-x^2-2xh-h^2] - [4+9x-x^2])/h`

`= lim_(hrarr0) (9h-2xh-h^2)/h`

`= lim_(hrarr0) (h(9-2x-h))/h`

`= lim_(hrarr0) (9-2x-h)`

`= 9-2x`

So `f'(x) = 9-2x`

Now the second derivative is the derivative of the first, so we have:

`f''(x) = lim_(hrarr0) (f'(x+h)-f'(x))/h`

`= lim_(hrarr0) ([9-2(x+h)] -[9-2(x)])/h`

`= lim_(hrarr0) ([9-2x-2h] -[9-2x])/h`

`= lim_(hrarr0) (-2h)/h`

`= -2`

So `f''(x) = -2`