Question

Use the limit definition of the derivative to calculate f'(x) for f(x) = x^2 -2

Answer 1

\[f'(x) = \frac{ f(x + \Delta x) - f(x) }{ \Delta x }\]

Answer 2

we can write this:
\[\frac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}} = \frac{{\left\{ {{{\left( {x + \Delta x} \right)}^2} - 2} \right\} - \left( {{x^2} - 2} \right)}}{{\Delta x}} = ...?\]

Answer 3

Now find f(x + dx):

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

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

(x+dx)^2 - x^2 / dx

x^2 + dx^2 + 2x*dx - x^2/dx

dx^2 + 2x*dx / dx = dx + 2x ( dx -> 0)
= 2x