Partial derivatives: Total derivatives

TOTAL DERIVATIVES

if u = f(x,y) where x = ϕ(t) and y = ψ(t) then we can express u as a function of t alone by substituting the values of x and y in f(x, y). Thus we can find the ordinary derivative du/dt which is called the total derivative of u to distinguish it from the partial derivatives ∂u/∂x and ∂u/∂y.

Now to find du/dt without actually substituting the values of x and y in f(x, y) we establish the following Chain rule:

du/dt = ∂u/∂x.dx/dt + ∂u/∂x.dy/dt      ... (i)

Proof. We have u = f(x,y)

Giving increment δt to t, let the corresponding increments of x,y and u be δx, δy and δu respectively. Then

u + δu = f(x + δx, y + δy)

Subtracting, δu = f(x + δx, y +δy) − f(x, y)

 = | f(x + δx, y+δy) − f(x,y+δy)| + |f(x,y+δy) ‒f(x,y) |

[Supposing ∂f(x,y)/∂x to be a continuous function of y]

 = [ ∂f(x,y)/∂x . dx/dt ] + [ ∂f(x,y)/∂y . dy/dt ] which is the desired formula.

Cor. Taking t=x, (i) becomes du/dx = ∂u/∂x + ∂u/∂y.dy/dx

Obs. If u = f(x, y, z), where x. y, z are all functions of a variables t, then chain rule is

Chain Rule for Total Derivatives

Let u= f(x, y, ...) be a continuous function of several variables x, y, .. with continuous partial derivatives ∂u/∂x, ∂u/∂y, …. If each variable is a function t, that is, x = x(t), y = y(t), and so on.

Then the total derivative of u with respect to t is given by

This is known as Chain Rule for total derivatives.

If u = f(x,y,z) be such that y and z are function of x. Then ƒ is a function of one independent variable. Then the total derivative of ƒ is given by

Composite functions

If u = f(x, y), where x=ϕ(t), y=ψ(t), then u is called a composite function of single variable t and we can find du/dt.

If u = f(x,y) where x = ϕ(r,s), y = ψ(r,s), then u is called a composite function of two variables u and v so that we can find du/dr and du/ds.

Differentiation of composite function:

Composite Function of One Variable

(1) If u is composite function of t, defined by the relations

 u = f(x,y); x = ϕ(t), ψ(t), then  ……(1)

The differential form (1) can be written as du = du/dx dx + du/dy dy, du is called the total differential of u.

If u = f(x, y, z) and x, y, z are functions of t, then

2) If u = f(x,y), where y = ϕ(x), then u is a composite function of x alone.

3) If we are given an implicit function f(x, y) = c, then u = f(x, y), where u = c.

Since, u = c, du/dx = 0

From (a), we get

By the similar way, we can prove

Composite Function of More Than One Variable

If u = f(r,s,t) and r, s, t are functions of x, y and z, then