What is this derivative? Confusing derivatives

Question

if $g(x)=y $ and $y'= \frac{\partial y}{\partial x}$

then what is this $\frac{\partial y'}{\partial y}$ ? would it be zero since $y'$ is not dependent on $y$? what if $y'$ is an implicit derivative, obtained from $f(x,y)=0$? what if $y'$ is defined in a differential equation such as $y'=h(x,y)$?

these kind of questions about partial and total derivatives make me very confused. I think I always get confused about the mathematical notation.

can you recommend me any books that could help me clear these notation problems I have?

Answer 1

No partial derivatives. By the chain rule, $$\frac{dy’}{dy} = \frac{dy’}{dx}\Big/\frac{dy}{dx} = \frac{y’’}{y’}.$$

Answer 2

I do not know any books about that, but I can recommend a few online resources, along with how I would answer it. Let's start with my explanation. So you start with the derivative chain rule, modified to fit in with partial derivatives;

$\frac{\partial y'}{\partial y}=\frac{\partial y''}{\partial y'}$

Using that, and the fact that $y=g(x)$, you just have to find the derivative of $g(x)$ (since the prime notation is the same as the $\frac{dx}{dy}$ notation, and yes, $y'$ is dependant upon $y$), and the equation from the beginning becomes $y'=\frac{\partial [g(x)]}{\partial x}$, which we've just simplified down to the partial derivative of $g(x)$.

Now for the resources:

1. https://www.khanacademy.org/math/in-in-grade-12-ncert/xd340c21e718214c5:continuity-differentiability/xd340c21e718214c5:chain-rule/a/chain-rule-overview

2. https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-1/a/derivative-notation-review

3. https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/partial-derivative-and-gradient-articles/a/introduction-to-partial-derivatives

Hopefully this was helpful in answering your questions about the confusing notations and other stuff that come along with learning differential calculus.

P.S.

It's a really hard part of learning calculus that you are struggling on. A few months ago, I was EXTREMELY confused with that very part of it. I know what it's like to feel overwhelmed by the extremities of this subject. It's really tough, but I assure you, that by the end of this month, you'll be a master at this.