Question

Find the derivative of the following function from the first principle.

x²

Answer 1

Let f(x) = x² then f(x + h) = (x + h)² 

Now `"d"/"dx"`f(x)

`= lim_(h->0) ("f"(x + "h") - "f"(x))/"h"`

`= lim_(h->0) ((x + "h")^2 - x^2)/"h"`

`= lim_(h->0) (x^2 + "h"^2 + 2"h"x - x^2)/"h"`

`= lim_(h->0) ("h"^2 + 2"h"x)/"h"`

`= lim_(h->0) ("h"("h" + 2x))/"h"`

`= lim_(h->0)` h + 2x

= 0 + 2x = 2x

Thus `"d"/"dx" (x^2)` = 2x