12. PARTIAL DIFFERENTIATION

In many mathematical and scientific problems, a function depends on more than one variable. To find how the function changes with respect to one variable while keeping others constant, we use partial differentiation.

Definition of Partial Derivative

If z = f(x, y), the partial derivative of z with respect to x is:

∂z/∂x = (∂/∂x) f(x, y)

Similarly, partial derivative with respect to y is:

∂z/∂y = (∂/∂y) f(x, y)

In partial differentiation, one variable is treated as constant while differentiating with respect to the other.

Notation Used

  • ∂f/∂x
  • fx
  • D₁f
  • fx₁

Higher-Order Partial Derivatives

Just like ordinary differentiation, we can differentiate again:

  • Second partial derivative w.r.t x:
    ∂²f/∂x²
  • Second partial derivative w.r.t y:
    ∂²f/∂y²
  • Mixed partial derivatives:
    ∂²f/∂x∂y or ∂²f/∂y∂x

Clairaut’s Theorem (Equality of Mixed Derivatives)

If f(x, y) is continuous and well-behaved, then:

∂²f/∂x∂y = ∂²f/∂y∂x

Important Rules of Partial Differentiation

1. Sum and Difference Rule

∂/∂x (u ± v) = ∂u/∂x ± ∂v/∂x

2. Product Rule

For z = uv:

∂z/∂x = u (∂v/∂x) + v (∂u/∂x)

3. Quotient Rule

For z = u/v:

∂z/∂x = (v ∂u/∂x − u ∂v/∂x) / v²

4. Chain Rule

If x and y depend on another variable t:

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

Homogeneous Functions

A function f(x, y) is called homogeneous of degree n if:

f(tx, ty) = tⁿ f(x, y)

Euler’s Theorem on Homogeneous Functions

If f is homogeneous of degree n, then:

x (∂f/∂x) + y (∂f/∂y) = n f(x, y)

Total Differentiation

If z = f(x, y), then total differential dz is:

dz = (∂f/∂x) dx + (∂f/∂y) dy

This is useful when both x and y change simultaneously.

Applications of Partial Differentiation

  • Finding maxima and minima of functions of two variables
  • Solving optimisation problems in economics and pharmacy
  • Heat flow, diffusion equations, and physical sciences
  • Rate of change where multiple variables contribute

Examples (Conceptual Overview)

  • Finding ∂z/∂x for z = x²y + 3xy²
  • Finding mixed partial derivatives for z = eˣ cos y
  • Checking homogeneity of functions
  • Using Euler’s theorem to verify degree of function
  • Using total differentiation for dz

Practice Questions

  • Find ∂z/∂x and ∂z/∂y for z = x²y + 5y³
  • Find ∂²z/∂x∂y for z = x³y²
  • Check if f(x, y) = x³ + y³ is homogeneous. If yes, find its degree.
  • Use Euler’s theorem for f(x, y) = x² + xy
  • Find dz if z = x²y and x, y change by small amounts dx, dy

Detailed Notes:

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