Partial derivatives

PARTIAL DERIVATIVES

A partial derivative of a function of several variables is the ordinary derivative w.r.to one of the variables when all the remaining variables are kept constant. Partial derivatives of higher order are obtained by partial differentiation of first order partial derivative.

FIRST ORDER PARTIAL DERIVATIVES

Let u = f(x, y) be a function of two independent variables. Differentiating u with respect to x keeping y as a constant is known as the partial differential coefficient of 'u' with respect to 'x'. It is denoted by ∂u/ ∂x.

Similarly, if we differentiate u with respect to 'y' keeping 'x' constant is known as

the partial differential coefficient of 'u' with respect to 'y'. It is denoted by

HIGHER ORDER PARTIAL DERIVATIVES

Let u= f(x,y) be a function of two variables x and y. Then ∂u/∂x and ∂u/∂y represent the first partial derivative of u with respect to x and y. Here both ∂u/∂x and ∂u/∂y again a function of x and y. Here each of these partial derivatives may again be differentiated with respect to x and y and it is denoted by

Note: ∂²u/∂x∂y means first we have to differentiate 'u' partially with respect to 'y' and then differentiate partially with respect to x.

1. If u = f(x, y) and its partial derivatives are continuous, then ∂²u/∂x∂y = ∂²u/∂y∂x.

2. If z = f(x), a function of independent variable x, then and ∂z/∂x and dx/dz represent the same.

3. If z is a function of two variables u and v, where u = ϕ(x,y) and v = ψ(x, y) then z is a function of x and y.

Example 1.

Find the first order partial derivative of u = tan^‒1[ x²+y² / x+y]

Solution: We have to find ∂u/∂x and ∂u/∂y.

Since u remains same if we interchange x and y in (1),

u is symmetrical w.r.to x and y.

Interchanging x and y in (2), we have

Similarly

Example 2. Find the first order partial derivatives of u = cos^‒1(x/y)

Solution: We have to find ∂u/∂x and ∂u/∂y.

Example 3. Find the first and second partial derivatives of z = x³+ y³ ‒ 3axy.

Solution: We have z = x³ + y³ ‒ 3axy

∂z/∂x = 3x²‒0‒3ay(1) = 3x² ‒ 3ay

and ∂z/∂y = 0+3y² ‒ 3ax(1) = 3y² – 3ax

∂²z/∂x² = ∂/∂x(3x²‒3ay) = 6x,

∂²z/∂y∂x =  ∂/∂y(3x² ‒ 3ay) = ‒3a,

∂²z/∂y² = ∂/∂y(3y² ‒ 3ax) = 6y

∂²z/∂x∂y = ∂/∂x(3y²‒3ax) = ‒3a,

We observe that ∂²z/∂y∂x = ∂²z/∂x∂y

Example 4.

Example 5. If u = log√[x² + y² + z²], then prove that (x² + y² + z²) (∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² ] = 1

Solution:

EXERCISE

39. If u = log(x² + y²) + tan^‒1(y/x), prove that ∂²u/∂x² + ∂²u/∂y² = 0.

40. Examine whether the following functions are homogenous or not

(i) x² + xy + y², (ii) x³cos(y/x)

41. If z(x + y) = x² + y², then show that (∂z/∂x ‒ ∂z/∂y)² = 4(1 ‒ ∂z/∂x ‒ ∂z/∂y)

42. If u = x²y + y²z + z²x, then prove that ∂u/∂x + ∂u/∂y + ∂u/∂z = (x + y + z)²

43. If x² + y² + z² − 2xy ‒ 2xyz = 1, then show that dx/√[1‒x²] + dx/√[1‒x²] + dx/√[1‒x²] = 0.

44. If u = (x − y)⁴ + (y‒z)⁴ + (z ‒ x)⁴, then show that ∂u/∂x + ∂u/∂y + ∂u/∂z = 0.

45. If u = log(cotx + coty + cotz), then prove that

sin 2x(∂u/∂x) + sin 2y(∂u/∂y)  + sin 2z(∂u/∂z) = ‒2