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  Differential Calculus, Physics tutorial

Partial Derivatives:

The partial derivative of the function of numerous variables is its derivative with respect to one of those variable with others held constant.

x = x(z,y)........................................Eq.1

From equation 1, x a dependent variable is the function of two independent variables z and y . Partial derivative of x with respect to y with z held constant is (∂x/∂y)z

For example, if

x = zy2........................................Eq.2

Then, partial derivative of x with respect to y with z held constant is

(∂x/∂y)z = 2zy........................................Eq.3

Likewise, partial derivative of x with respect to z with y held constant is

(∂x/∂z)y = y2........................................Eq.4

Exact Differential:

Assume that there exists the relation among three coordinates x, y, and z in such a way that x is a function of y and z (i.e. x(z, y) ); therefore

f(x, y, z) = 0........................................Eq.5

Exact differential of x (dx) is

dx(∂x/∂y)z dy + (∂x/∂z)ydz........................................Eq.6

Usually for any three variables x , y , and z we have relation of form

dx = M(y,z)dy + N(y,z)dz........................................Eq.7

If differential dx is exact, then

(∂M/∂z)y = (∂N/∂y)z........................................Eq.8

Implicit Differential:

Consider the equation of form

xy = x2y2........................................Eq.9

One can differentiate two sides of equation 9 using equation 6 (i.e differentiating both the left and right hands side with respect to x while y is held constant and with respect to y while x is held constant).

(∂(xy)/∂x)ydx + (∂(xy)/∂y)xdy = (∂(x2y2)/∂x)ydx + (∂(x2y2)/∂y)xdy........................................Eq.10

Equation 10 gives

ydx + xdy = 2xy2dx + 2x2ydy........................................Eq.11

Collecting like term and then factorize to have

dy/dx = (2xy2 - y)/(x - 2x2y) ........................................Eq.12

Consider equation 9 as f = xy - x2 y2 (i.e. moving expression in right side of equation 9 to left side and then equate result to f ). Then

dy/dx = -(∂f/∂x)/(∂f/∂y) ........................................Eq.13

Product of Three Partial Derivatives:

Assume that there exists the relation among three coordinates x, y, and z; therefore

f(x, y, z) = 0........................................Eq.14

Then x can be imagined as the function of y and z

dx = (∂x/∂y)zdy + (∂x/∂z)ydz........................................Eq.15

Also y can be imagined as the function of x and z, and

dy = (∂y/∂x)zdx + (∂y/∂z)xdz........................................Eq.16

Insert equation 16 in 15

dx = (∂x/∂y)z[(∂y/∂x)zdx + (∂y/∂z)xdz] + (∂x/∂z)ydz

Rearrange to get:

dx = (∂x/∂y)z(∂y/∂x)zdx + [(∂x/∂y)z(∂y/∂z)x + (∂x/∂z)y]dz........................................Eq.17

If dz = 0 dx ≠ 0 it follows that

(∂x/∂y)z(∂y/∂x)z = 1

(∂x/∂y)z = 1/((∂y/∂x)z)........................................Eq.18

In eq.17 if dx = 0 and dz ≠ 0, it follows that:

(∂x/∂y)z(∂y/∂z)x + (∂x/∂z)y = 0

Move (∂x/∂z)y to other side of equation to get

(∂x/∂y)z(∂y/∂z)x = -(∂x/∂z)y........................................Eq.19

Then divide both sides of equation 19 by (∂z/∂x)y

(∂x/∂y)z(∂y/∂z)x(∂z/∂x)y = -1........................................Eq.20

This is known as minus-one product rule.

Chain Rule of Partial Derivatives:

Another helpful relation is known as chain rule of partial derivatives. Assume T is function of V and P, and that each of V and P is the function of Z, then

(∂T/∂V)P = (∂T/∂Z)P(∂Z/∂V)P........................................Eq.21

Equation 21 is chain rule of partial derivative. The following can as well be written:

(∂S/∂P)T = (∂S/∂V)T(∂V/∂P)T........................................Eq.22(a)

(∂U/∂V)P = (∂U/∂T)P(∂T/∂V)P........................................Eq.22(b)

Equation 21 and 22 are known as chain rule of partial derivatives

Second Derivatives or Second Order Derivatives:

Let f (x, y) be function with continuous order derivatives, then we can compute first derivatives to be (∂f/∂x)z and (∂f/∂z)x. One can further compute second derivatives ∂2f/∂x2, ∂2f/∂z2 ∂2f/∂x∂y, and ∂2f/∂y∂x. Take note of these two second order derivates i.e. ∂2f/∂x∂y and ∂2f/∂y∂x, they are known as mixed second derivatives. It can be shown that mixed second derivatives are equal, i.e. it doesn't matter order will perform differentiation.

∂2f/∂x∂y = ∂2f/∂y∂x

Functions of More than Two Variables:

Assume that f (x, y, z), derivative of f with respect to one of the variables with other two constant (e.g. derivative of f x with y and z constant) can be written as:

(∂f/∂x)yz, (∂f/∂y)xz, and (∂f/∂z)xy

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