Differential Equations

Question

I do not understand the concept of differential in physics for example

What have both the sides been differentiated by to get this result.I understand that $dz^2/dz$ gives $2z$, where has the extra $dz$ come from? Also, what has the side with $\theta$ been differentiated by, to get this result? It cannot be $dz$. If it's differentiated with respect to $\theta$, how can you differentiate something with respect to another thing in two sides of an equation? Can someone explain this and how did $dz$ and $d\theta$ come in the equations?

Answer 1

It is not extra $dz$ - it is the differential, as opposed to a derivative. Differentials and derivatives are different things, although they are interchangeable in the case of a function of one variable, where $$df(z) = f'(z)dz$$ (df is the differential of $f$, $dz$ is the differential pf $z$, $f'$ is the derivative of $f$). For a function of many variables $$df(x_1, x_2,..., x_n) = \sum_{i=1}^n\frac{\partial f(x_1, x_2,..., x_n)}{\partial x_i}dx_i,$$ where df is the differential, whereas the expansion coefficients are called partial derivatives.