When is differentiating an equation valid?

I wonder that Is it true to differentiate an equation side by side. Under which conditions can I differentiate both sides. For example, for the simple equality $x=3$, Is ıt valid to differentiate both sides with respect to x. I know that I am missing some basic point but I cant find it. Thanks for your helps.

If you are given that for all $x$, $f(x)=g(x)$, then the two functions are equal, and so their derivatives must be as well. Therefore, $f'(x)=g'(x)$ for all $x$.

On the other hand, if you are trying to find a solution to $f(x)=g(x)$, differentiating may not retain truth. Consider, for all $x$, $f(x)=2$ and $g(x)=1$. Clearly, $f(x)=g(x)$ has no solutions, but $f'(x)=g'(x)$ has infinitely many solutions.

In your example, $x=3$, the equation is not true for all $x$. It is true for only one $x$, that is, $3$. Because of this, differentiating both sides can lead to a false statement.

Without getting into the details of the definition of function, let us just note that the two most important underlying concepts in the concept of function are its domain and its assignment rule (with a bucket of salt).

To say that two functions $f,g$ are equal is to say that the assignment rule is the same and that the domains are the same.

Differentiating is something that you do to functions, so when you talk about 'differentiating $x=3$' this can only make sense if you look at it as meaning 'differentiating both sides of the equality $f=g$ where $f\colon A\to \mathbb R, x\mapsto x$ and $g\colon A\to \mathbb R, x\mapsto 3$, for some set open set $A$'. Note that $f=g$ means $\forall x\in A(f(x)=g(x))$

Now you can differentiate both sides of $f=g$ to obtain $f'=g'$ or $\forall x\in A(1=0)$.

There is no contradiction here because the initial assumption $f=g$ is false, it is not true that $\forall x\in A(f(x)=g(x))$. (Note that $A$ is open so this excludes the possbility of $A=\{3\}$).