Quick Summary
Differentiation finds the instantaneous rate of change of a function. This chapter builds from the first principles definition to all standard methods: product rule, quotient rule, chain rule, implicit differentiation, parametric differentiation, and logarithmic differentiation. These tools power the applications chapter (Chapter 7 of Class 12).
Definition — First Principles
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
This is the slope of the tangent line at x.
| Differentiability implies continuity — but continuity does NOT imply differentiability (e.g., | x | is continuous at 0 but not differentiable there). |
|---|
Standard Derivatives
| Function | Derivative |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| aˣ | aˣ ln a |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| cot x | −cosec²x |
| sec x | sec x tan x |
| cosec x | −cosec x cot x |
| sin⁻¹x | 1/√(1−x²) |
| cos⁻¹x | −1/√(1−x²) |
| tan⁻¹x | 1/(1+x²) |
Differentiation Rules
| Rule | Formula |
|---|---|
| Sum/Difference | (f ± g)' = f' ± g' |
| Product rule | (fg)' = f'g + fg' |
| Quotient rule | (f/g)' = (f'g − fg') / g² |
| Chain rule | dy/dx = (dy/du)(du/dx) |
| Scalar | (kf)' = kf' |
Chain Rule
If y = f(u) and u = g(x): dy/dx = (dy/du) × (du/dx)
Example: y = sin(x²) → dy/dx = cos(x²) × 2x = 2x cos(x²)
Implicit Differentiation
When y is defined implicitly (e.g., x² + y² = r²):
1. Differentiate both sides with respect to x. 2. Apply chain rule to y-terms (multiply by dy/dx). 3. Solve for dy/dx.
Example: x² + y² = 25 2x + 2y(dy/dx) = 0 → dy/dx = −x/y
Parametric Differentiation
If x = f(t) and y = g(t):
dy/dx = (dy/dt) / (dx/dt)
Example: x = at², y = 2at → dx/dt = 2at, dy/dt = 2a → dy/dx = 1/t
Logarithmic Differentiation
Used for yˣ, xˣ, or complex products/quotients.
Steps: Take ln of both sides → differentiate implicitly → solve for dy/dx.
Example: y = xˣ → ln y = x ln x (1/y)(dy/dx) = ln x + 1 → dy/dx = xˣ(1 + ln x)
Worked Examples
Example 1: Find d/dx [x³ sin x]. Product rule: 3x² sin x + x³ cos x
Example 2: Find d/dx [(x²+1)/(x−1)]. Quotient rule: [(2x)(x−1) − (x²+1)(1)] / (x−1)² = (x² − 2x − 1)/(x−1)²
Example 3: Differentiate y = sin(e^(3x)). Chain rule: dy/dx = cos(e^(3x)) × e^(3x) × 3 = 3e^(3x) cos(e^(3x))
Example 4: Find dy/dx if xy + y² = 2. Differentiate: y + x(dy/dx) + 2y(dy/dx) = 0 dy/dx(x + 2y) = −y → dy/dx = −y/(x + 2y)
Important Questions
One Mark
- Find d/dx [tan⁻¹(x)].
- Differentiate e^(sin x) with respect to x.
- State the product rule of differentiation.
- Find dy/dx if y = ln(cos x).
Short Answer (3 marks)
- Differentiate √(x + 1/√x) with respect to x.
- If x = a cos θ, y = a sin θ, find dy/dx.
- Find dy/dx using implicit differentiation: x³ + y³ = 3axy.
Long Answer (5 marks)
- From first principles, find the derivative of sin x.
- If y = (sin x)^(cos x), find dy/dx using logarithmic differentiation.
Common Mistakes
- Product rule: writing (fg)' = f'g' instead of f'g + fg'.
- Quotient rule: wrong sign — it is (f'g − fg')/g², NOT (fg' − f'g)/g².
- Chain rule: forgetting to multiply by the derivative of the inner function.
- Implicit differentiation: not writing dy/dx when differentiating a y-term — every d(y)/dx needs the dy/dx factor.
Board Exam Tips
- First principles derivation of d/dx(sin x) or d/dx(xⁿ) is a 5-mark board question — know the full proof.
- Implicit differentiation questions like "find dy/dx for x³ + y³ = 3axy" appear every year as 3-mark questions.
- Logarithmic differentiation for y = xˣ or y = (sin x)^(tan x) is a standard 3-mark question.
- JEE: Higher-order derivatives, Leibniz theorem, and parametric differentiation appear frequently.