Question

Why the derivative of (x^2/a^2) = (2x/a^2)? ​

Answer 1

I assume you're referring to a function,

$f(x) = \dfrac{x^2}{a^2}$

where a is some unknown constant. By definition of the derivative,

$\displaystyle f'(x) = \lim_{h\to0}{f(x+h)-f(x)}h$

Then

$\displaystyle f'(x) = \lim_{h\to0}{\frac{(x+h)^2}{a^2}-\frac{x^2}{a^2}}h \\\\ f'(x) = \frac1{a^2} \lim_{h\to0}{(x+h)^2-x^2}h \\\\ f'(x) = \frac1{a^2} \lim_{h\to0}{(x^2+2xh+h^2)-x^2}h \\\\ f'(x) = \frac1{a^2} \lim_{h\to0}{2xh+h^2}h \\\\ f'(x) = \frac1{a^2} \lim_{h\to0}(2x+h) = \boxed{\frac{2x}{a^2}}$