Overview
This chapter introduces differentials (for approximation and errors) and partial derivatives of functions of several variables, including Euler's theorem.
1. Differentials and Approximation
dy = f'(x) dx
f(x + Δx) ≈ f(x) + f'(x)Δx
Relative error = dy/y , Percentage error = (dy/y) × 100
2. Partial Derivatives
∂f/∂x : differentiate w.r.t. x, treat y as constant
∂f/∂y : differentiate w.r.t. y, treat x as constant
Higher order: ∂²f/∂x², ∂²f/∂y², and mixed ∂²f/∂x∂y = ∂²f/∂y∂x
3. Euler's Theorem (homogeneous functions)
If f(x, y) is homogeneous of degree n, then:
x(∂f/∂x) + y(∂f/∂y) = n·f
Worked Examples
Example 1 — Approximation
Find √26 approximately using differentials.
f(x) = √x, x = 25, dx = 1, f'(x) = 1/(2√x) = 1/10
√26 ≈ √25 + (1/10)(1) = 5 + 0.1 = 5.1
Example 2 — Error estimation
The side of a cube is measured as 10 cm with error 0.1 cm. Find the approximate error in volume.
V = x³ → dV = 3x² dx = 3(100)(0.1) = 30 cm³
Example 3 — Partial derivatives
For f = x²y + 3xy², find ∂f/∂x and ∂f/∂y.
∂f/∂x = 2xy + 3y²
∂f/∂y = x² + 6xy
Example 4 — Euler's theorem
Verify Euler's theorem for f = x² + xy + y² (degree 2).
∂f/∂x = 2x + y , ∂f/∂y = x + 2y
x(2x+y) + y(x+2y) = 2x² + xy + xy + 2y² = 2(x² + xy + y²) = 2f ✓
Common Mistakes
- Forgetting dx in dy = f'(x)dx.
- Not holding the other variable constant in partial differentiation.
- Misjudging the degree of homogeneity in Euler's theorem.
- Confusing absolute error (dy) with percentage error.
Exam Tips & Revision Checklist
- Use a nearby exact value (25, 64, …) for differential approximations.
- ∂/∂x treats y as a number; differentiate as usual.
- Practised: approximation, error in volume, partial derivatives, Euler's theorem.
Frequently Asked Questions
dy = f'(x) dx approximates the small change in y for a small change dx in x; it is used for approximations and error estimation.