Determine f'(3) from first principles if f(x)=-5x^2.

We have to find f'(3) for f(x)=-5x^2, using the first
principle.

Now from the first
principle,

f'(3) = lim [f(x) - f(3)]/(x-3), for
x->3

=> f'(3) = lim (-5x^2 + 45)/(x - 3), for
x->3

=> f'(3) = lim [(-5)(x^2 - 9)]/(x-3),
for x->3

=> f'(3) = lim [(-5)(x-3)(x
+3)/(x-3), for x->3

=> f'(3) = (-5)*lim
(x+3), for x->3

=> f'(3) = -5*(3
+3)

=> f'(3) =
-5*6

=> f'(3) =
-30

The required value of f'(3) is
-30