Question

Use the definition of the derviative to compute the derivative of f(x)= 1- 7x^2 at the specific point x=2.

Answer 1

`f'(x) = -28 \ at \ x = 2`

Explanation:

`f'(2) = \lim_(h \to 0)(f(2+h)) -f(2))/(h) \\\\f(2+h) = 1 - 7(2+h)^2 = 1 - 7(4 +h^2 +4h) = 1 - 28 - 7h^2 - 28h = - 7h^2 -28h - 27\\\\f(2) = 1 - 7(2^2) = 1 - 28 = -27\\\\f(2+h) - f(2) = -7h^2 - 28h - 27 - (-27) = -7h^2 -28h -27 + 27 = -7h^2 -28h\\\\f'(2) = \lim_(h \to 0)(-7h^2 - 28h)/(h) \\\\`

`= \lim_(h \to 0) (h(-7h -28))/(h)\\\\= \lim_(h \to 0) (-7h -28)}\\\\= -7 (0) - 28\\\\= -28`