Gosh, they are everywhere. A derivative of a function of one variable expresses a change in that function relative to its argument. For example Newtons law for rectilinear motion is F = mass times the second derivative of distance with respect to time.
But the world involves functions that depend on several variables. For example the pressure of a gas depends on density and temperature. The speed of sound (squared), it turns out, in a nebula in space (which is very nearly at constant temperature due to radiative transport) is the partial derivative of the pressure with respect to density keeping temperature fixed.
Your happiness H depends on how much money, m, you make and the number of hours, h, you spend with your family. H = H(m, h). But how much money you make also depends on how much on how much time you spend with your family. The more time you spend with them the less money you will make. So m = m(h) and we must write
H = H(m(h), h)
Now, we want to know how many hours "h" to work to maximize happiness so we take the _total_ derivative of H with respect to h and set it equal to zero:
Equations involving partial derivatives are known as partial differential equations (PDEs) and most equations of physics are PDEs:
(1) Maxwell's equations of electromagnetism
(2) Einstein's general relativity equation for the curvature of space-time given mass-energy-momentum.
(3) The equation for heat conduction (Fourier)
(4) The equation for the gravitational potential of a blob of mass (Newton-Laplace)
(5) The equations of motion of a fluid (gas or liquid) (Euler-Navier-Stokes)
(6) The Schrodinger equation of quantum mechanics
(7) The Dirac equation of quantum mechanics
(8) The Yang-Mills equation
(9) The Liouville equation of statistical mechanics
Please, someone mention use of partial derivatives or at least functions of several variables that anyone might use in their daily work in a regular job as engineer or technician or in some type of analytical work/decision making. One vague idea I have in mind is physical behavior of blended materials: Their composition and temperature endurance and flow properties and other physical values. Some independant variables might not truely be independant of other independant variables, possibly making any multivariable function confusing.