If you're content to just "look it up" on everything, yes, damn near everything has been derived at one point or another, and all you really need is a search engine, or a good solid data book.
But if you're the kind of person who needs to understand things, and wants to see theorems proven in class, and wants to derive things yourself, you will need it.
Personally, I use arithmetic daily; algebra frequently, during design (most often playing with resistor dividers, AC steady state filters, diodes and transistors, etc.); and calculus and diff eq occasionally (more at complicated coupling networks, time-domain stuff, certain models of reality).
I use Fourier and E&M (except for special cases for transformers) extremely rarely, however they are always intuitively present in my mind. Two cases where manual calculation is almost useless, but knowledge and intuition with their properties is invaluable. Practical calculations are almost exclusively computed, for example DFT on time series data, or FEA on E&M.
The most recent diff eq I looked at was this problem:
Suppose you have a heat source in the center of a round metal plate. The heat source has a radius r1 and the plate, r2. Thickness and conductivity do not matter directly, but the product, the heat spreading coefficient rho, does. The surface has a thermal conductivity to the surroundings defined as p = T(r) * k, where p is power density (W/m^2 or whatever), T(r) is the temperature at radius r (0 <= r <= r2) and k is the heat transfer coefficient. (This is at best a poor approximation of radiation -- which goes as Tabs^4, and a poor approximation of convection, which is nonlinear in the T^2 range or so, depending on airflow and awful stuff like that.) So, a transistor bolted to the chassis and how it heats up, simplified.
Setting up the equations is fairly straightforward, the trouble comes when trying to solve it... it turns out to be of the form of the Bessel function. So you kind of have to compute it, or pick it out of a table or whatever, to do much with it.
Another thermal conductivity problem I've done is uniform power (W/m^2) into the face of a conductor, which sinks the heat laterally (no surface dissipation or crap like that to deal with). It's a differential equation again, but easy to solve this time; it's a parabola, with the vertex at the hot end. The temperature at the end is simply the sum of all temp rises along the conductor (which, at a given point, is the sum of all power dissipated by that point, which is linearly cumulative).
Tim