if anyone out there uses calculus derivatives in their career please let me know how!?

Question

please include what it is you do (your job/career)...
and how calculus deritvatives are used

Answer 1

Electronic engineering - circuit analysis:

Solving a circuit for the current and voltage according to Kirchhoff's Laws and these device equations:
Resistor : v = R * i
Capacitor: i = C * dv/dt
Inductor : v = L * di/dt

Anything complicated you generally solve by using the Laplace transform, in which case the above equations simplify to
for V,I = Laplace transforms of v,i:
Resistor : V = RI
Capacitor: I = sCV
Inductor : V = sL

Then you solve the system of equations for V,I and apply a reverse Laplace transform.

Answer 2

Astrophysicist, First Officer of the Starship Enterprise

(My doctorate was always ignored in your earthly television series, because it would have confused me with your own Dr Spock, who wrote the most widely used book used in child rearing at that time. Of course there was really a joke here: my rôle in the series was in fact child-sitting your terrestrian astronauts as they took their first baby steps "to boldly go where no MAN has gone before." [The series also failed to emphasize that we Vulcans had often been there long earlier, although there were implicit references to that from time to time when Vulcan influences were noted in the farthest regions of the Universe.])

Example A: Spaceflight trajectories

At much less than warp speed, the free fall motion of the Enterprise is governed by the equation:

m x dv/dt = local gravitational force due to the vector sum of all gravitational attraction, where m is the mass of the Enterprise.

(When using fuel, it's a little more complicated, as m is then changing. This means that the rate of change of momentum --- what your terrestrian Newton's 2nd law of motion really deals with --- contains an additional term, namely v x dm/dt.)

Example B: The structure and evolution of stars

When studying stellar evolution, the (changing) structure of stars is governed both by four internal spatial differential equations and also temporal derivatives. (Thus this really involves partial derivatives.) If time changes are slow enough (the "quasi-static approximation") one can get away with just the four spatial derivative equations.

It would take us too far afield to give them all, but the "first two equations" are basic and fairly obvious:

1. How the local radial change in the mass M_r contained inside an internal sphere of radius r is related to the local density, rho:

dM_r/dr = 4 x pi x r^2 x rho (1)

2. How radial changes in the local pressure P(r) depend upon M_r, the mass interior to r, and local density rho:

dP/dr = - G x M_r x rho / r^2 (2)

(By Newton's celebrated result on the internal attraction of a sphere, the strength of the local internal gravitational acceleration, g(r) is responsible for the combination - G x M_r / r^2 that multiplies rho on the r.h.s. above; the "-" indicating that [vector!] g(r) "points inward, of course.)

You'll notice that eqn (2) introduces a new variable, P, so you've got one new eqn. and one new variable. It doesn't seem like we're getting anywhere!

The "third eqn." (which I'm not going to attempt to write here) tells you how the local "temperature gradient" depends on energy transport. This introduces yet another new variable, T, the temperature, but also the energy flow. Does this mean we've got two new variables but only one new equation connecting them? (HELP! Are we going backwards? Well, not necessarily. See what follows.)

IF we have an "equation of state" connecting P, rho and T, then the "new" T isn't really new, because you can view it as a (known) function of P and rho. (In practice, things are really the other way round: P and T can be regarded as "basic" --- because that's convenient with the direct equations #2 and #3 available for their derivatives --- and rho can be regarded as a "consequence" of P and T.) So what is really new is the amount of energy that's flowing locally. (Astrophysicists call the FULL amount of energy flowing out of the internal sphere of radius r the "luminosity at r," L_r. THAT really IS a new variable. The local energy flow per unit area is then L_r/[4 x pi x r^2], and that combination, with other stuff, is what appears in this "third equation.")

So, the next obvious question is: what produces the energy that's flowing? (Actually, I've jumped a step here. When there is a temperature gradient inside a star, energy will necessarily flow: it's inevitable, as it "flows from hot to cold," and all that. So, the point I jumped to logically was: "If energy is necessarily flowing but the system is assumed to be quasi-static, what's ultimately providing more 'new' energy to make up for what is being continually lost from a given region?")

It looks as though we're going deeper and deeper into the mire: IS THIS SYSTEM OF EQUATIONS AND VARIABLES EVER GOING TO CLOSE?!

Yes, at this stage it can. Let the rate at which the energy flow L_r is itself changing with r inside the star be provided (in the quasi-static approximation) by nuclear reactions. They have a certain rate at which they are locally "PRODUCING" their energy. Well, guess what?: the reaction rates can be expressed as local functions of the (known) composition, T, and rho. BINGO! The equations are finally "closed": in loose terms, we "have as many independent variables as equations for them." That means that we are finally equipped to solve them.

This is probably much more than you needed or indeed wanted to know, but we astrophysicists just couldn't do our work without using calculus derivatives.

Live long and prosper.