Question

Find the derivatives of the following functions using first principle.

f(x) = – x² + 2

Answer 1

f(x + Δx) = – (x + Δx)² + 2

f(x + Δx) – f(x) = – [x² + 2x Δx + (Δx)²] + 2 – [– x² + 2]

`(f(x + Deltax) - f(x))/(Deltax) = (- x^2 + 2xDeltax - (Deltax)^2 + 2 + x^2 - 2)/(Deltax)`

`(f(x + Deltax) - f(x))/(Deltax) = (- 2xDeltax - (Deltax)^2)/(Deltax)`

`(f(x + Deltax) - f(x))/(Deltax) = (-2xDeltax)/(Deltax) - (Deltax)^2/(Deltax)`

`(f(x + Deltax) - f(x))/(Deltax) = - 2x - Deltax`

`lim_(Deltax -> 0) (f(x + Deltax) - f(x))/(Deltax) = lim_(Deltax -> 0) (- 2x) - lim_(Deltax -> 0) Deltax`

`f"'"(x) = - 2x - 0`

`f"'"(x) = - 2x`