Total Derivative, Gradient, Divergence, and Curl | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Partial and Total Derivatives

Partial derivatives of first order 
Let z = f(x, y) be a function of two independent variables x and y. If y is kept constant and x alone is allowed to vary then z becomes a function of x only. The derivative of z with respect to x, treating y as constant, is called partial derivative of z with respect to x and is denoted by
Similarly the derivative of z with respect to y, treating x as constant, is called partial derivative of z with respect to y and is denoted by

Thus
 are called first order partial derivatives of z

Partial derivatives of higher order

In general,

Solved Examples 1: First order partial derivative of u = y^x is
Solution : u = y^x
Treating y as constant

treating x as constant

Euler’s theorem on Homogenous functions

Adding (1) and (2) we get

Note: Euler’s theorem can be extended to a homogenous function of any number of variables. Thus if u is a homogenous function of degree n in x, y and z then

Solved Examples 2: If  show that .
Solution: 

f (x, y) is a homogenous function of degree -2 in x and y
By Euler’s theorem, we have

Composite functions

(ii) If z = f(x, y) where x = ϕ(u, v), y = Ψ(u, v)
then z is called a composite function of (2 variables) u & v so that we can find

Total derivative of composite functions

Solved Example 3: Find du/dt when u = x when u = xy² + x² y, x = at², y = 2at.
Solution: The given equations define u as a composite function of t.

= 2a³ t³ (5t + 8)
Also u = xy² + x² y = at² . 4a² t² + a² t⁴ . 2at = 4a³ t⁴ + 2a³ t⁵