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Roo2

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## Homework Statement

I'm reviewing physics using Feynman's Lectures, and I'm finding that he frequently uses implicit differentiation in his lessons. This is unfortunate for me because I never got the hang of it beyond the simplest cases. I'm currently going through the proof that the gravitational potential inside a hollow sphere is zero. I almost got through it, but at one point (outlined below in red) he uses an implicit derivative and loses me. Could someone please explain how this works?

I'm used to two-variable implicit differentiation, where one variable (y) is dependent upon the independent variable (x). Thus, if you have an implicitly defined function such as

x^2 + y(x)^2 = 25

it can be solved by deriving with respect to this independent variable:

d(x^2)/dx + d(y^2)/dx = d(25)/dx

2x + 2y*dy/dx = 0

dy/dx = -x / y

Is this an incorrect way of thinking about it? I can't relate it to what Feynman is doing in the image above. First, he doesn't seem to be deriving with respect to anything; he's just taking the differential. What are the rules for this operation? Second, it's as if he's deriving with respect to r on the left side (r^2 ---> 2r dr) and with respect to x on the right side (a^2 + R^2 -2Rx ---> -2R dx). How is it "mathematically legal" to arbitrarily pick different variables to differentiate with respect to on the two sides of the equation?

On a hopefully more simple note, in the blue box within the red box, where did he hide the negative sign which appears in front of 2R dx?

Thanks to anyone who can help clear this up. I never really got implicit differentiation down in high school calculus and somehow I got through multivariate and diff eq without really understanding the concept.

## Homework Equations

d/dx (F(y(x)) = dF/dy * dy/dx

## The Attempt at a Solution

Stated above