Find The Partial Derivative With Respect To X, Y, Z), Derivatives 101: What Does With Respect To Mean

So I got to the implicit differentiation section in the Calc book and while I managed to understand how to do the exercises, I don't have any intuitive understanding as to what “differentiating with respect to” means. The book completely ignored any conceptual understanding of the idea and just offered some basic guidelines for doing implicit differentiation.

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Let's say you have an equation that looks something like y = x2 + sin(x) + 7, and you want to know “How much does y change when I change x a little bit?”. To do this, you take the derivative of the expression, and x is the variable that you are differentiating. That is, you are differentiating with respect to the variable x. It just means that you care about changing x a little bit to see what happens with y.

The same thing is happening in implicit differentiation, except you don't always have nice equations like y = x2 + sin(x) + 7. For example, the equation of a circle (radius 1 centered at the origin) is x2 + y2 = 1. We still want to know how does y change when we change x. We can see that in a picture. If you draw out a circle and put your pencil down and move it along the circle in the x-direction, you'll see that at the top and bottom of the circle, y doesn't change very much when x changes, but at the left and right of the circle, a tiny change in x corresponds to a huge change in y.

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This is what we are trying to do with implicit differentiation. It's just a technique to find derivatives of these curves that don't have nice function representatives. (In fact, the circle curve isn't even a function!) The terminology “with respect to x” just means that x is the variable that we are changing, and we want to see how y reacts to changes in x.

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We could also do it the other way. Going back to that circle example — what if we wanted to see how much x changes if we did a little change in y? So instead of moving x a little and seeing how y reacts, we want to move y a little and see how x reacts. How would we do this? Well, we would use implicit differentiation again, except the variable of differentiation would be x, not y. (Mentally, we can just swap the roles of x and y, and it's the same thing). Then instead of getting a dy/dx at the end, we would get dx/dy, and we would say that we differentiated “with respect to y”.

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