Paper I — Mathematics

Differentiation

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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High yield5 mapped NDA questions

Build the idea

Differentiation measures instantaneous change. NDA questions reward recognising the outermost operation: products need the product rule, quotients the quotient rule, and nested functions the chain rule. Simplify algebraically before differentiating whenever possible.

Function, tangent and accumulated area

Concept

Differentiation gives the instantaneous rate of change (slope) of a function. The standard rules — power, product, quotient and chain — let you differentiate almost any combination of functions.

Mastering the rules makes calculus questions quick and mechanical.

Key Points & Formulas

d/dx(xⁿ) = n x^(n−1); d/dx(sinx)=cosx; d/dx(eˣ)=eˣ; d/dx(lnx)=1/x.
Product rule: (uv)' = u'v + uv'.
Quotient rule: (u/v)' = (u'v − uv')/v².
Chain rule: dy/dx = (dy/du)(du/dx).

Worked Example

Q. Differentiate y = x³ ln x with respect to x.

Product rule with u = x³, v = ln x.

dy/dx = (3x²)(ln x) + (x³)(1/x).

= 3x² ln x + x².

Common traps

Forgetting the inner derivative in the chain rule.
Applying the product rule to a constant multiple.
Writing d(ln x)/dx = ln x/x instead of 1/x.

NDA shortcuts

Rewrite roots and reciprocals as powers.
Logarithmic differentiation is fastest for variable powers and large products.
Memorise derivatives in inverse pairs with common integrals.

One-screen revision sheet

Recall before you practise

d(xⁿ)/dx = nxⁿ⁻¹
(uv)'=u'v+uv'
d[f(g(x))]/dx=f'(g(x))g'(x)

Past-paper bank

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5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA Practice 2026

$\dfrac{d}{dx}\big(e^{2x}\big)$ equals

Question 2

NDA I 2025

What is $\lim\limits_{x \to 0} f'(x)$ equal to?

Question 3

NDA I 2024

Consider the following statements:
1. f(x) is increasing in (e, ∞)
2. f(x) is decreasing in (1, e)
3. 9ln7 > 7ln9
Which of the statements given above are correct?

Question 4

NDA I 2024

The non-negative values of b for which the function (16x³/3 − 4bx² + x) has neither maximum nor minimum in the range x > 0 is

Question 5

NDA I 2024

Consider the following statements:
1. f''(e) = 1/e
2. f(x) attains local minimum value at x = e
3. A local minimum value of f(x) is e
Which of the statements given above are correct?

← Limits, Continuity and Differentiability Continue in Test Module Application of Derivatives (Tangents, Maxima & Minima) →