calc

How would I go about finding the derivative of x^(1/2) using first principles?

1/2 - 1 =
1/2 - 2/2 =
-1/2

d/dx x^(1/2) = 1/2(1/sqrt(x))

>>2
That's not first principle, that's using a derived result.

consider \frac{\sqrt(x) - \sqrt(x_0)}{x-x_0}
= \frac{\sqrt(x) - \sqrt(x_0)}{x-x_0}*\frac{\sqrt(x) + \sqrt(x_0)}{\sqrt(x) + \sqrt(x_0)}
= \frac{x - x_0}{(x-x_0)(\sqrt(x) + \sqrt(x_0))}
= \frac{1}{\sqrt(x) + \sqrt(x_0)}
(of course, if you were smarter than I am, you could have just factored the denominator from the start, and skipped several steps)

The derivative is the limit of this thing as x -> x₀

Alternatively, one could use the binomial expansion, but then you'd have to prove that the (infinite) binomial expansion actually converges to sqrt(x).

test

2^²2