An ordinary derivative from single-variable calculus asks a simple question: if the one input to a function changes slightly, how does the output change? Many real quantities, though, depend on several variables at once — the volume of a gas depends on both its temperature and its pressure, for instance, and both can change independently. A partial derivative extends the idea of a derivative to exactly this situation, by isolating how the function responds to a change in just one of its several variables, while every other variable is deliberately held fixed, as though frozen, for the purposes of that particular calculation.
Freezing the other variables is a deliberate simplification, not an error
Holding the other variables constant might sound like it's ignoring real complexity in the underlying situation, but it's actually a deliberate and genuinely useful simplification, not a mistake or an approximation glossed over. By asking "how does the output change if only this one variable moves, with everything else held exactly fixed," a partial derivative gives a precise, well-defined answer to a genuinely meaningful question — how sensitive the function actually is to that one specific variable, isolated from whatever the other variables happen to be doing at the same time. A function of several variables can have a separate partial derivative for each one of its variables, and together, that full set of partial derivatives captures how the function responds to each individual input separately, providing the building blocks used to understand how the function behaves when several variables change simultaneously as well.
Reassembling the full picture from the isolated pieces
Once you have every variable's partial derivative in hand, they can be recombined to describe what happens to the function's output when several variables change at the same time, not just one in isolation — a tool called the total differential does exactly this, adding up each variable's individual contribution, weighted by how much that particular variable actually changed, to approximate the function's overall change. This two-step approach — isolate each variable's individual effect first, then recombine those effects to understand the fuller, simultaneous picture — is what makes multivariable calculus tractable at all for problems that would otherwise require reasoning about several interacting variables changing all at once, a genuinely difficult thing to do directly without first breaking it down into these individually manageable pieces.
What we're still unsure about
Partial derivatives and the total differential are rigorously defined, well-established mathematical tools, with no genuine dispute over how they're calculated or what they formally represent. What does require real care and judgement in applied settings, rather than being purely a matter of following a fixed procedure, is deciding which variables actually matter enough to include explicitly in a given real-world multivariable model in the first place — a genuinely complex physical, economic or biological system typically depends on far more variables than anyone could practically track individually, and choosing which ones to model explicitly and which to fold into a simplified constant is a modelling decision, not something the mathematics of partial differentiation itself can settle for you.
This sits inside Multivariable Calculus & Partial Derivatives, one of eight topics in Calculus, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.