Definition: A partial derivative: a derivative with respect to one independent variable of a function in multiple variables while holding the other variables constant.
Alright class, settle down, let’s take a look, At ‘partial,’ a word that needs a little hook! The dictionary gives us a helpful plea, "A partial derivative… for you and me!" Let's break it down with a rhythmic flow, Think of functions where numbers grow. Imagine a function, say f(x, y) , you see, With two variables – x and y, agree? Now, when we find a derivative’s grace, We focus on one variable's space. We hold the others steady, nice and tight, Like holding a candle in the fading light. So if we take the derivative of f with respect to x , While keeping y constant, that's what we expect! And if we do the same for y , you see, Holding x steady – perfectly free! That’s a ‘partial,’ my dears, it’s clear and true, A single change, just focusing through. In short: It's like taking a peek at how one thing changes when everything else stays the same. It’s about isolating a specific variable. Do you have any questions? Don’t be shy, let’s try!