How to get the derivative of a physical formula?

Question

I've heard that if $U_{ind} \neq $ constant, you must use the following formula:

$ U_{ind} = N \times \dfrac{d \phi}{dt}$

I however don't know how to get the derivative of this formula? In math I basically have a formula like $x^2-2x$ and the derivative is just $2x - 2$, but how would this work with a physics formula?

My main problem is that $\phi$ and $t$ are just numbers, and the derivative of a number is 0..

Answer 1

First, ϕ and t are not just numbers, they are both variables, and in this particular example ϕ is a function of t. But what is shown in that physical equation is a given law, or a definition, such as v=dx/dt. You cannot use it as is unless you have either an explicit or an implicit dependence for a particular problem. For instance, find the velocity v at time t=5 of an object moving attached to a spring whose equation is x=7*sin(4*t+2). In your specific question, they have to give you ϕ(t), otherwise you cannot use the equation to compute $U_{ind}$.

Answer 2

If $N$ is constant, then $ \dfrac{\mathrm d U_{ind}}{\mathrm dt} = N \times \dfrac{\mathrm d^2 \phi}{\mathrm dt^2}$.