How do you find the derivative of -5x^2+8x+2 using the limit definition?

1 Answer
Sep 5, 2016

8 - 10 x

Explanation:

We have: f(x) = - 5 x^(2) + 8 x + 2

Let's evaluate the derivative of this expression by differentiating using first principles:

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

=> f'(x) = lim_(h -> 0) (((- 5 (x + h)^(2) + 8 (x + h) + 2) - (- 5 x^(2) + 8 x + 2)) / (h))

=> f'(x) = lim_(h -> 0) ((- 5 (x^(2) + 2 h x + h^(2)) + 8 x + 8 h + 2 + 5 x^(2) - 8 x - 2) / (h))

=> f'(x) = lim_(h -> 0) ((- 5 x^(2) - 10 h x - 5 h^(2) + 8 h + 5 x^(2)) / (h))

=> f'(x) = lim_(h -> 0) ((8 h - 10 h x - 5 h^(2)) / (h))

=> f'(x) = lim_(h -> 0) (8 - 10 x - 5 h)

=>f'(x) = 8 - 10 x