Continuity, Differentiability and Applications of Derivatives
Why This Chapter Matters
Derivatives account for 15-20 marks in CBSE Class 12 — the single largest topic. Continuity, differentiability, chain rule, implicit differentiation, tangent/normal, and maxima/minima are all tested.
Core Concepts
1. Continuity
f(x) is continuous at x = a if:
Every polynomial, rational (where defined), and trigonometric function is continuous.
2. Differentiability
f is differentiable at x=a if f'(a) = lim(h→0) [f(a+h)-f(a)]/h exists.
Differentiable → Continuous. Continuous does NOT imply differentiable.
|x| is continuous at x=0 but NOT differentiable (sharp corner).
3. Standard Derivatives
d/dx(xⁿ) = nxⁿ⁻¹ | d/dx(eˣ) = eˣ | d/dx(aˣ) = aˣ ln a
d/dx(ln x) = 1/x | d/dx(log_a x) = 1/(x ln a)
d/dx(sin x) = cos x | d/dx(cos x) = -sin x | d/dx(tan x) = sec²x
d/dx(sin⁻¹ x) = 1/√(1-x²) | d/dx(cos⁻¹ x) = -1/√(1-x²)
d/dx(tan⁻¹ x) = 1/(1+x²) | d/dx(cot⁻¹ x) = -1/(1+x²)
4. Rules of Differentiation
Product rule: (uv)' = u'v + uv'
Quotient rule: (u/v)' = (u'v - uv')/v²
Chain rule: d/dx[f(g(x))] = f'(g(x)) × g'(x)
Logarithmic differentiation: For y = f(x)^g(x) or complex products.
Take ln both sides → differentiate → multiply back by y.
Implicit differentiation: When y is not explicitly isolated.
Differentiate both sides with respect to x, treat y as function of x.
d/dx(y²) = 2y·dy/dx.
Parametric: x=f(t), y=g(t). dy/dx = (dy/dt)/(dx/dt).
5. Second Derivative
y'' or d²y/dx² = d/dx(dy/dx).
Second derivative test for extrema:
If f'(a)=0 and f''(a) < 0 → local maximum.
If f'(a)=0 and f''(a) > 0 → local minimum.
If f'(a)=0 and f''(a) = 0 → inconclusive.
6. Applications of Derivatives
Tangent and Normal:
Slope of tangent at (x₀,y₀) = f'(x₀).
Equation of tangent: y - y₀ = f'(x₀)(x - x₀).
Slope of normal = -1/f'(x₀).
Rate of change: dy/dt = (dy/dx)·(dx/dt).
Increasing/Decreasing:
f'(x) > 0 on (a,b) → f increasing on (a,b).
f'(x) < 0 on (a,b) → f decreasing.
Maxima/Minima (First Derivative Test):
f'(x) changes + to - at x=a → local max.
f'(x) changes - to + at x=a → local min.
Approximation: f(x+Δx) ≈ f(x) + f'(x)·Δx. (Using differentials: Δy ≈ dy = f'(x)dx)
Board Examples
Q1: Find dy/dx if y = sin(x²).
Chain rule: dy/dx = cos(x²) × 2x = 2x cos(x²).
Q2: Find the equation of tangent to y = x³ - 3x + 2 at (1, 0).
dy/dx = 3x²-3. At x=1: slope = 0. Horizontal tangent: y-0=0(x-1) → y=0.
Q3: A ladder 10m long leans against a wall. Top slides down at 2 m/s. How fast is foot moving out when top is 6m above ground?
x²+y²=100. 2x(dx/dt)+2y(dy/dt)=0. At y=6: x=8.
dx/dt = -y(dy/dt)/x = -6(-2)/8 = 1.5 m/s.
PYQs (CBSE)
CBSE 2023: Find local maxima and minima of f(x)=x³-6x²+9x+15.
f'(x)=3x²-12x+9=3(x²-4x+3)=3(x-1)(x-3). Critical points: x=1,3.
f''(x)=6x-12. f''(1)=-6<0 → local max at x=1. f''(3)=6>0 → local min at x=3.
f(1)=1-6+9+15=19 (local max). f(3)=27-54+27+15=15 (local min).
CBSE 2022: Differentiate sin⁻¹(2x√(1-x²)) with respect to x.
Let x=sinθ → 2x√(1-x²)=2sinθcosθ=sin2θ.
y=sin⁻¹(sin2θ)=2θ=2sin⁻¹x. dy/dx=2/√(1-x²).