Partial Derivatives – Differential Calculus for GATE, ESE, AE/JE

Partial Derivatives are a fundamental concept in Differential Calculus for functions of multiple variables. This topic is very important for GATE, ESE, AE/JE exams and forms the base for advanced topics like Euler’s Theorem and Maxima & Minima (Two Variables).

Basic Concept

If a function depends on more than one variable, then the derivative taken with respect to one variable while keeping others constant is called a partial derivative.

If z = f(x, y), then:
∂z/∂x = partial derivative w.r.t x
∂z/∂y = partial derivative w.r.t y

First Order Partial Derivatives

Example:

z = x²y + y³

∂z/∂x = 2xy
∂z/∂y = x² + 3y²

Second Order Partial Derivatives

∂²z/∂x², ∂²z/∂y², ∂²z/∂x∂y

Mixed derivatives:

∂²z/∂x∂y = ∂²z/∂y∂x

(For continuous functions — Clairaut’s Theorem)

Important Results

• Order of differentiation does not matter (if function is continuous)
• Notation: fₓ, fᵧ, fₓₓ, fₓᵧ
• Used in optimization and multivariable calculus

Chain Rule

If z = f(x, y), where x and y depend on t:

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

Total Differential

For small changes:

dz = (∂z/∂x)dx + (∂z/∂y)dy

Solved Examples

Example 1

z = x²y + y²

∂z/∂x = 2xy
∂z/∂y = x² + 2y

Example 2

z = x² + y² + xy

∂²z/∂x∂y = 1
∂²z/∂y∂x = 1

Equal (verified)

Example 3 (Chain Rule)

z = x² + y², x = t, y = t²

dz/dt = 2x(dx/dt) + 2y(dy/dt)

= 2t(1) + 2t²(2t) = 2t + 4t³

Example 4

If z = xy, then ∂²z/∂x∂y = ?

Answer: 1

Shortcut Tricks

• Treat other variables as constant
• Use symmetry of mixed derivatives
• Practice chain rule carefully
• Avoid sign mistakes

Questions

Q1. Order of mixed derivative depends on?

Answer: Continuity

Q2. ∂/∂x (y²)?

Answer: 0

Exam Tips

• Don’t differentiate all variables together
• Focus on mixed derivatives
• Practice chain rule problems
• Use standard notation properly

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