# What is this derivative? Confusing derivatives

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?

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

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: