The derivative of $ x $ in an interval $ [a,b] $ on which a function $ f $ is defined is defined as..
$$f'(x)=\lim_{t \to x}\frac{f(t)-f(x)}{t-x}$$
Why is this equal to
$$ f'(t)=\lim_{x \to t}\frac{f(x)-f(t)}{x-t}?$$
2026-04-26 13:20:45.1777209645
Equivalence of two definitions of the derivative of a real function
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The $t$ and $x$ are just dummy variables, or placeholders. It doesn't matter what symbols you use to represent the variables:
$$\displaystyle \rho ' (\circ) := \lim_{\star \to \circ}\frac {\rho(\star) - \rho(\circ)}{\star-\circ}$$
It is usually best to use the same variable in the definition of the function as in the limitand. So when we previously defined $\rho$, we might have had $\rho(\star) = \text{something}$. But that's not necessary, and it usually doesn't matter.