Question

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

Answer 1

I assume you're referring to a function,

`f(x) = (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){((x+h)^2)/(a^2)-(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{(2x)/(a^2)}`