Product rule in calculus This is wonderful question I came across whiles doing calculus. We all know that $$\frac{d(AB)}{dt} = B\frac{dA}{dt} + A\frac{dB}{dt}.$$ 
Now if $A=B$ give an example for which 
$$\frac{dA^2}
{dt} \neq 2A\frac{dA}{at}.$$ 
I have tried many examples and could't get an example, any help?

let's observe function 
$y=(f(x))^2$ , this function can be decomposed as the composite of two functions:
$y=f(u)=u^2$ and $u=f(x)$
So :
$\frac { d y}{ d u}=(u^2)'_u=2u=2f(x)$
$\frac{du}{dx}=f'(x)$
By the chain rule we know that :
$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}=2f(x)f'(x)$
