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Question
Find the derivative of the following function from the first principle.
x2
Solution
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
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