Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
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.