Differentiate y = x^2 using the first principle.

To differentiate y = x^2 using the first principle, we need to find the limit of the difference quotient as h approaches 0.

The difference quotient is given by:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

For y = x^2, we have:

f(x) = x^2
f(x + h) = (x + h)^2 = x^2 + 2hx + h^2

Now, substitute these values into the difference quotient formula:

f'(x) = lim(h->0) [(x + h)^2 - x^2] / h
f'(x) = lim(h->0) [x^2 + 2hx + h^2 - x^2] / h
f'(x) = lim(h->0) [2hx + h^2] / h
f'(x) = lim(h->0) 2x + h
f'(x) = 2x

Therefore, the derivative of y = x^2 using the first principle is f'(x) = 2x.

Using the first principle, the derivative of y = x^2 is given by:

dy/dx = lim[h->0] (f(x+h) - f(x))/h
= lim[h->0] ((x+h)^2 - x^2)/h
= lim[h->0] (x^2 + 2xh + h^2 - x^2)/h
= lim[h->0] (2xh + h^2)/h
= lim[h->0] 2x + h
= 2x

Therefore, the derivative of y = x^2 is dy/dx = 2x.