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when function is defined ?

function is defined as

A function is an equation for which any x that can be plugged into the equation will yield exactly one y out of the equation.

where we used partial and where diffentiate?

quations 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

So you see PDEs are fundamental.

Solution:  Let’s first take the derivative with respect to x and remember that as we do so all the y’s will be treated as constants.  The partial derivative with respect to x is,

Notice that the second and the third term differentiate to zero in this case. It should be clear why the third term differentiated to zero. It’s a constant and we know that constants always differentiate to zero. This is also the reason that the second term differentiated to zero. Remember that since we are differentiating with respect to x here we are going to treat all y’s as constants. That means that terms that only involve y’s will be treated as constants and hence will differentiate to zero.

Now, let’s take the derivative with respect to y. In this case we treat all x’s as constants and so the first term involves only x’s and so will differentiate to zero, just as the third term will. Here is the partial derivative with respect to y.

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