Differential Calculus Questions for NDA

Limits and Continuity

A limit asks: what value does f(x) approach as x gets close to some point a? You do not care what happens AT x=a, only near it. A function is continuous at x=a if the left-hand limit equals the right-hand limit and both equal f(a). Think of continuity as being able to draw the graph without lifting your pen.

• LHL = RHL = f(a) is the three-part condition for continuity
• Removable discontinuity: hole in graph, limit exists but does not equal f(a)
• Jump discontinuity: LHL and RHL both exist but are not equal
• Infinite discontinuity: function blows up to infinity at that point
• Polynomial and trig functions are continuous everywhere in their domain
• At x=a, always check 0/0 or infinity/infinity forms before substituting

Differentiability and Derivatives of Standard Functions

A function is differentiable at a point if the derivative exists there — meaning the graph has a unique tangent (no sharp corner, no vertical tangent). Every differentiable function is continuous, but NOT vice versa. Memorise the derivatives of standard functions cold — they appear in nearly every question.

• If differentiable then continuous — converse is NOT always true (classic NDA trap)
• |x| is continuous at 0 but NOT differentiable at 0 — sharp corner
• d/dx(x^n) = n*x^(n-1) for all real n
• d/dx(e^x) = e^x, d/dx(ln x) = 1/x
• d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec^2 x
• d/dx(sec x) = sec x tan x, d/dx(cosec x) = -cosec x cot x, d/dx(cot x) = -cosec^2 x

Implicit Differentiation and Parametric Differentiation

When y is NOT explicitly written as a function of x (like x^2 + y^2 = r^2), differentiate both sides with respect to x and remember that d/dx(y) = dy/dx using chain rule. In parametric form, x and y are both given in terms of a third variable t. Then dy/dx = (dy/dt) divided by (dx/dt).

• Implicit: differentiate both sides, collect dy/dx terms on one side
• Parametric: dy/dx = (dy/dt) / (dx/dt)
• Second derivative in parametric: d2y/dx2 = d/dt(dy/dx) divided by (dx/dt)
• Common parametric pair: x = a cos t, y = a sin t (circle)

Applications — Tangents, Normals, and Rate of Change

The derivative dy/dx at a point gives the slope of the tangent at that point. The normal is perpendicular to the tangent. Rate of change questions ask you to find how fast one quantity changes with respect to time — just differentiate with respect to t.

• Slope of tangent at (x1, y1) = dy/dx evaluated at that point
• Slope of normal = -1 / (slope of tangent)
• Equation of tangent: y - y1 = m(x - x1) where m = dy/dx at that point
• If dy/dx = 0, tangent is horizontal; if dx/dy = 0, tangent is vertical
• Rate of change: express relation, differentiate both sides with respect to t

Maxima and Minima

At a maximum or minimum, the first derivative is zero — these are called critical points. Use the second derivative test to classify: if f''(x) < 0 at a critical point, it is a local maximum (curve bends down like a hill). If f''(x) > 0, it is a local minimum (bends up like a valley). NDA frequently gives word problems asking you to maximize area or minimize cost — set up the function, differentiate, solve.

• Step 1: Find f'(x) = 0 to get critical points
• Step 2: Find f''(x) at each critical point
• f''(x) < 0 means maximum (hill), f''(x) > 0 means minimum (valley)
• If f''(x) = 0, use higher derivatives or first-derivative sign change test
• Global max/min on closed interval: check critical points AND endpoints
• Increasing function: f'(x) > 0; Decreasing function: f'(x) < 0

• Assuming continuity implies differentiability — it does NOT. |x| at x=0 is the classic counter-example that NDA has used directly.
• Forgetting the chain rule when differentiating composite functions like sin(x^2) or e^(3x) — writing just cos(x^2) instead of 2x cos(x^2) loses marks.
• In quotient rule, writing (u'v + uv') instead of (v*u' - u*v') / v^2 — mixing it up with product rule under exam pressure.
• In maxima-minima word problems, finding the critical point but forgetting to verify it is actually a maximum or minimum using the second derivative test.
• In parametric differentiation, computing d(dy/dt)/d(dx/dt) directly for second derivative instead of the correct formula: differentiate dy/dx with respect to t, then divide by dx/dt.

• HILL or VALLEY trick for second derivative: Negative second derivative = Hill = Maximum. Positive second derivative = Valley = Minimum. Remember N-H-V-P: Negative-Hill, Valley-Positive.
• Product Rule as UV formula: d(UV) = U dV + V dU. Remember it as Each takes a turn being differentiated.
• Quotient Rule: LO dHI minus HI dLO, over LO squared — where LO = denominator, HI = numerator.
• Chain rule: Peel the onion — differentiate the outermost layer first, keep the inner function intact, then multiply by the derivative of the inner layer.
• Differentiable implies Continuous but NOT the other way: D is a subset of C. Think D for Diamond inside a Circle C.

• NDA papers almost always have 1-2 questions on standard limits — especially sin(x)/x type and (1+x)^(1/x) = e type. Memorise these cold, no derivation needed.
• Maxima-minima application problems (maximize area of rectangle, minimize surface area of cylinder) appear every 1-2 papers. Practice setting up the function from the word problem — that is the actual skill being tested.
• Tangent and normal questions are straightforward mark-scorers. They are formulaic: find dy/dx, plug in point, write line equation. Never skip these in the exam.
• Implicit differentiation of circle or ellipse equations appears regularly. x^2 + y^2 = r^2 and x^2/a^2 + y^2/b^2 = 1 are the two most common curves.
• NDA does NOT require epsilon-delta proofs of limits. All limit questions are computation-based. Focus on L-Hopital, factoring, and standard limit formulas — not theory.