Find the derivative of the function using the definition of derivative. state the domain of the function and the domain of its derivative f(x)=4+8x-5x^2

Question

Please help me 

Answer 1

First, what is the definition of the derivative?  It is:

 

f'(x) = lim_h->0 (f(x+h)-f(x))/h)

(Conceptually, this is just the average slope over a distance h, as h goes to zero)

 

Plugging in our specific equation, this gives

 

(4 + 8(x+h) - 5(x^2+2xh+h^2) - (4 + 8x - 5x^2))/h

= (8x + 8h - 5x^2 - 10xh - 5h^2 - 8x + 5x^2)/h

= (8h - 10xh - 5h^2)/h

= 8 - 10x - 5h

Which, as h goes to 0, is just

8 - 10x

 

Both the original function and its derivative will yield a valid output for any x you use, so both domains are (-∞,∞)