Differential Calculus Class 10 Notes
A concise formula sheet covering key differentiation and integration concepts.
Differential Calculus — Detailed Notes
Differential Calculus is an important chapter in Mathematics and is frequently tested in both conceptual and application-based questions. Students should first understand the core definition, then connect the topic with real-life observations and exam patterns.
Differential calculus studies the rate of change of a function — the derivative represents the instantaneous rate of change or the slope of the tangent at a point. In school and entrance exams, questions usually check your conceptual clarity, step-wise logic, and ability to avoid common mistakes.
To prepare effectively, break Differential Calculus into smaller sub-parts: definition, laws/rules, examples, formulas, and revision questions. After theory, solve short questions, then move to mixed-level numericals or application prompts.
A smart revision strategy is to maintain a one-page summary for Differential Calculus. Include important terms, two solved examples, and last-minute checkpoints before exams.
Key Exam Points
- Derivative definition: f'(x) = lim[h→0] (f(x+h)-f(x))/h; geometric meaning: slope of tangent.
- Rules: power (d/dx xⁿ = nxⁿ⁻¹), product (uv)' = u'v+uv', quotient (u/v)' = (u'v-uv')/v², chain.
- Standard derivatives: d/dx(sin x)=cos x, d/dx(eˣ)=eˣ, d/dx(ln x)=1/x.
- Maxima/minima: f'(x)=0 (critical point); f''(x)<0 → max; f''(x)>0 → min.
- Mean Value Theorem: f'(c) = (f(b)-f(a))/(b-a) for some c in (a,b).
- Implicit differentiation: differentiate both sides w.r.t. x, apply chain rule for y terms.
Important Formula / Rule
d/dx(xⁿ)=nxⁿ⁻¹ | Chain: dy/dx=(dy/du)·(du/dx) | For tangent at (x₀,y₀): y-y₀=f'(x₀)(x-x₀)
What You Will Learn in Differential Calculus
Differential calculus studies the rate of change of a function — the derivative represents the instantaneous rate of change or the slope of the tangent at a point.
- Derivative definition: f'(x) = lim[h→0] (f(x+h)-f(x))/h; geometric meaning: slope of tangent.
- Rules: power (d/dx xⁿ = nxⁿ⁻¹), product (uv)' = u'v+uv', quotient (u/v)' = (u'v-uv')/v², chain.
- Standard derivatives: d/dx(sin x)=cos x, d/dx(eˣ)=eˣ, d/dx(ln x)=1/x.
- Maxima/minima: f'(x)=0 (critical point); f''(x)<0 → max; f''(x)>0 → min.
- Mean Value Theorem: f'(c) = (f(b)-f(a))/(b-a) for some c in (a,b).
- Implicit differentiation: differentiate both sides w.r.t. x, apply chain rule for y terms.
Key Formulas
d/dx(xⁿ)=nxⁿ⁻¹Chain: dy/dx=(dy/du)·(du/dx)For tangent at (x₀,y₀): y-y₀=f'(x₀)(x-x₀)
Example
If f(x)=x³-6x²+9x, then f'(x)=3x²-12x+9=0 gives critical points at x=1 and x=3.
Solved Numerical Example
Find maximum and minimum of f(x) = 2x³ - 3x² - 12x + 4. f'(x) = 6x² - 6x - 12 = 6(x²-x-2) = 6(x-2)(x+1) = 0 → x=2 or x=-1. f''(x) = 12x-6. At x=2: f''=18>0 → local min, f(2)=-16. At x=-1: f''=-18<0 → local max, f(-1)=11.
Expected Exam Questions — Differential Calculus
Q1.Find the derivative of f(x) = x⁴ - 3x² + 2x - 7.
Q2.Find the equation of tangent to y=x²+3x at x=1.
Q3.A ladder 5m long leans against wall. If base slides out at 1 m/s, how fast is the top sliding down when base is 3m from wall?
🔘 MCQ Practice — Differential Calculus
MCQ 1.The derivative of sin(x²) is:
✓ Correct Answer: 2x·cos(x²)
MCQ 2.At a critical point where f'(x)=0 and f''(x)>0, the function has:
✓ Correct Answer: Local minimum
📥 Topic Materials
1 resource curated by SII educators for Differential Calculus
Calculus: Differentiation & Integration Formula Sheet
A concise formula sheet covering key differentiation and integration concepts.
🧭 Learning Roadmap
Follow this step-by-step path to master Differential Calculus
🎯 Step 2 — Core Concept
Deep understanding and theory