in depth can you explain how to use the defn. of a derivative to calculate the derivative of f(x) = -x^2+2?

Question

this is a test question, and if you could send me the workings that would be great.

Answer 1

nb2718 is almost correct except for one thing:

d/dx (-x^2 + 2) = -2x Where did that minus sign go?? That'd cost you 5 points on a test, or it would have with my calculus teacher!

f(x + h) = -(x +h)^2 + 2 = -[x^2 + 2xh + h^2]+2

f(x) = (-x^2 + 2)

lim h-->0 {[-x^2 - 2xh -h^2] + 2 - [(-x^2 + 2)]}/h =

lim h-->0 {-x^2 - 2xh -h^2 + 2 + x^2 - 2}/h =

lim h-->0 {h(-2x + h)/h} now divide by h before it goes to zero...

lim h--> {-2x + h} = -2x

QED

Answer 2

by definition, the derivative is (f(x+h) - f(x))/h as h approaches 0. So just substitute:

f(x+h) = (x+h)² + 2 = x² +2xh + h² +2, so

((x² +2xh + h² + 2) - (x² + 2))/h = (2xh + h²)/h

= 2x + h. And the limit of that as h goes to 0 is 2x.

Answer 3

The derivative of a function is defined based on how the function changes with respect to x, and this change is represented by its gradient as x changes.

f(x) = (-x)^2 + 2
f'(x) = -2x