I'm learning about implicit differentiation and i'm stuck at understanding the concept of "differentiating an equation".
If I define $ f(x) = x^2$ and $g_1(y) = 9$ and $g_2(y) = 9 + y^2$
Then I define 2 equations:
$(1)$ $f(x) = g_1(y)$ (i.e. $x^2 = 9$)
Then I can't differentiate both side of this equation with respect to x (ortherwise I will get: $2x = 0$)
$(2)$ $f(x) = g_2(y)$ (i.e. $x^2 = 9 + y^2$)
Then I can differentiate both side of this equation with respect to $x$ to get: $ 2.x = 2.y.\frac{dy}{dx}$
Could you please explain me the difference between the 2 cases ? And in general, in which condition could I differentiate both side of an equation concerning more than 1 variable ?
Thank you very much for your help!
Best Answer
$x^2=9$ is an equation that has a finite number of real solutions ($2$) for $x$. Differentiating this would not make much sense.
An equation like $x^2-1 = (x-1)(x+1)$ holds true for all real $x$. You can differentiate this and both sides will turn out to be equal.
In general, you can differentiate both sides of the equation $f_1(x,y)=f_2(x,y)$ iff this equation holds true for all real $x,y$ in a certain interval of the real numbers, containing infinitely many points.