📐Mathematics
Differential Calculus
Limits, continuity, differentiability, derivatives, applications — maxima, minima, tangents, normals.
Class 10Class 11Class 12
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.
Answer: f'(x) = 4x³ - 6x + 2.
Q2.Find the equation of tangent to y=x²+3x at x=1.
Answer: y(1)=4; y'=2x+3, y'(1)=5. Tangent: y-4=5(x-1) → y=5x-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?
Answer: x²+y²=25. Diff: 2x(dx/dt)+2y(dy/dt)=0. At x=3: y=4. dy/dt = -x/y × dx/dt = -3/4 × 1 = -0.75 m/s (sliding down at 0.75 m/s).
🔘 MCQ Practice — Differential Calculus
MCQ 1.The derivative of sin(x²) is:
A. cos(x²)
B. 2x·cos(x²) ✓
C. 2x·cos(2x)
D. cos(2x)
✓ Correct Answer: 2x·cos(x²)
MCQ 2.At a critical point where f'(x)=0 and f''(x)>0, the function has:
A. Local maximum
B. Local minimum ✓
C. Inflection point
D. Not enough information
✓ Correct Answer: Local minimum
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