I need somebody to explain me why
$\frac{d}{dx}(y^2)=2y\frac{dy}{dx}$
I don't get it. I'd say the left hand side is 0, because there is no change in $y^2$ when x changes slightly.
I need somebody to explain me why
$\frac{d}{dx}(y^2)=2y\frac{dy}{dx}$
I don't get it. I'd say the left hand side is 0, because there is no change in $y^2$ when x changes slightly.
Actually, the answer depends on what $y$ is. If $y$ is just a free variable, then you're right and the derivative is $0$. But if $y$ is a function of $x$ (i.e., $y=y(x)$) then the identity you wrote is an example of the chain rule in action.
You are considering that $y$ is some function of $x$. An example would be if $y=x^3$. Then $y^2=x^6, \frac{d}{dx}(y^2)=2y\frac{dy}{dx}=2x^3(3x^2)=6x^5=\frac{d}{dx}x^6$. This is an example of the chain rule in action.