NDA Maths · Free notes

Differentiation — NDA Mathematics

Differentiation is a high-volume NDA chapter — around 89 past-year questions across 2017–2026, and a prerequisite for Application of Derivatives, Limits & Continuity, and much of the calculus that follows. Most marks are won by recognising which TOOL a problem wants: a standard derivative, the chain rule, logarithmic differentiation for variable exponents, or a simplify-first trick on a messy inverse-trig expression. Work the three notes in order — first the core techniques (standard derivatives, the rules, chain and logarithmic differentiation), then the advanced forms (parametric, implicit, and higher-order derivatives), and finally differentiability itself (when the derivative exists at all — corners, the modulus, and the greatest-integer function). The traps are predictable: forgetting to convert degrees to radians, mishandling a power tower, or assuming a continuous function must be differentiable.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter's formulas and traps alone, on one page.

All 89 past questions NDA Maths strategy NDA home All formulas on one page

Subtopic notes

Formula & revision sheet

14 formulas · 1 reference tables · 17 gotchas across all subtopics — the exam-eve cheat-sheet

Core Techniques — Standard Derivatives, Rules, Chain & Logarithmic

Formulas (8)

The derivative as a limit (first principles) · First-principles definition $f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$
Product and quotient rules · Product and quotient rules $(uv)' = u'v + uv', \qquad \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
The chain rule (composite functions) · Chain rule $\frac{d}{dx}\,f(g(x)) = f'(g(x))\cdot g'(x)$
Logarithmic differentiation · Logarithmic differentiation $y = f(x)^{g(x)} \;\Rightarrow\; \frac{1}{y}\frac{dy}{dx} = g'(x)\ln f(x) + g(x)\frac{f'(x)}{f(x)}$
Derivative of one function with respect to another · Derivative of u w.r.t. v $\frac{du}{dv} = \frac{du/dx}{dv/dx}$
Simplify the inverse-trig first, then differentiate · Standard inverse-trig collapses $\tan^{-1}\dfrac{2x}{1-x^2} = 2\tan^{-1}x \qquad \sin^{-1}\dfrac{2x}{1+x^2} = 2\tan^{-1}x \qquad \cos^{-1}\dfrac{1-x^2}{1+x^2} = 2\tan^{-1}x \qquad \cos^{-1}(\sin x) = \dfrac{\pi}{2} - x \qquad \tan^{-1}\dfrac{a-b}{1+ab} = \tan^{-1}a - \tan^{-1}b$
Differentiating functional equations · Exponential functional equation $f(x+y)=f(x)f(y) \;\Rightarrow\; f'(x) = f'(0)\,f(x)$
Trigonometric Simplification Toolkit · The collapses you reach for most $1+\cos x = 2\cos^2\tfrac{x}{2},\quad 1-\cos x = 2\sin^2\tfrac{x}{2},\quad \sqrt{1\pm\sin 2x}=|\sin x\pm\cos x|$

Reference tables (1)

Standard derivatives to memorise18 rows

Function f(x)	Derivative f′(x)
$x^n$	$n\,x^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\sec x$	$\sec x\tan x$
$e^x$	$e^x$
$a^x$	$a^x\ln a$ The $\ln a$ factor is the most-forgotten part of the table.
$\ln x$	$\dfrac{1}{x}$
$\log_a x$	$\dfrac{1}{x\ln a}$
$\sin^{-1} x$	$\dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$	$\dfrac{1}{1+x^2}$
$\cot x$	$-\csc^2 x$
$\csc x$	$-\csc x\cot x$ The co-functions (cos, cot, cosec) all carry a MINUS sign.
$\cos^{-1} x$	$-\dfrac{1}{\sqrt{1-x^2}}$
$\cot^{-1} x$	$-\dfrac{1}{1+x^2}$
$\sec^{-1} x$	$\dfrac{1}{\|x\|\sqrt{x^2-1}}$
$\ln\|x\|$	$\dfrac{1}{x}$ for all $x\neq0$
$\tfrac{x}{2}\sqrt{x^2-a^2}-\tfrac{a^2}{2}\ln\big\|x+\sqrt{x^2-a^2}\big\|$	$\sqrt{x^2-a^2}$ The antiderivative of √(x²−a²), read backwards — a 'standard form' the PYQs quote without proof; the +a² sibling differentiates to √(x²+a²).

Radians only. The chain rule extends each of these to a composite argument.

Watch out for (10)

Degrees must be converted to radians first
→ Standard derivatives to memorise
Don't power-rule an exponential
→ Standard derivatives to memorise
Derivative of $\ln x$ is $1/x$ , not $\ln x$ or $x$
→ Standard derivatives to memorise
The product rule is not the product of derivatives
→ Product and quotient rules
Quotient rule — order and sign in the numerator matter
→ Product and quotient rules
Don't forget the derivative of the inner function
→ The chain rule (composite functions)
Don't quotient-rule the raw inverse-trig
→ Simplify the inverse-trig first, then differentiate
$1\pm\cos x$ (half-angle) vs $1\pm\cos 2x$ (power-reduction)
→ Trigonometric Simplification Toolkit
The root of a perfect square is a MODULUS
→ Trigonometric Simplification Toolkit
$\sec\pm\tan$ — mind which way the half-angle shifts
→ Trigonometric Simplification Toolkit

Parametric, Implicit & Higher-Order Derivatives

Formulas (4)

Implicit differentiation · Implicit rule of thumb $\frac{d}{dx}\big[\,y\text{-term}\,\big] = (\text{its derivative})\cdot\frac{dy}{dx}$
Parametric differentiation · Parametric first derivative $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
Higher-order derivatives · Second derivative $\frac{d^2y}{dx^2} = \frac{d}{dx}\!\left(\frac{dy}{dx}\right)$
Second derivative of an inverse — the d²x/dy² identity · Second derivative of the inverse $\frac{d^2x}{dy^2} = -\frac{d^2y/dx^2}{\left(dy/dx\right)^{3}}$

Watch out for (4)

Every $y$ -term needs a $\frac{dy}{dx}$ factor
→ Implicit differentiation
Don't invert the parametric ratio
→ Parametric differentiation
$\frac{d^2y}{dx^2}$ is not $\left(\frac{dy}{dx}\right)^2$
→ Higher-order derivatives
Second derivatives don't invert like first derivatives
→ Second derivative of an inverse — the d²x/dy² identity

Differentiability — When the Derivative Exists

Formulas (2)

The left-hand = right-hand derivative test · One-sided derivative test $f'(c^-) = f'(c^+) \iff f \text{ differentiable at } c$
Derivative at an awkward point via the limit definition · Derivative from first principles $f'(c) = \lim_{h \to 0} \dfrac{f(c+h) - f(c)}{h}$

Watch out for (3)

The implication only runs one way
→ Differentiable implies continuous (not the converse)
Continuity first, then slopes
→ The left-hand = right-hand derivative test
Not every |·| means non-differentiable
→ Modulus corners — and the x|x| trap

PYQ weightage by concept

21 concepts · 89 PYQs — where the marks actually sit, so you know what to drill first

Core Techniques — Standard Derivatives, Rules, Chain & Logarithmic51 PYQs · 57%

Concept	PYQs	Share
The derivative as a limit (first principles)	10	11%
The chain rule (composite functions)	9	10%
Standard derivatives to memorise	7	8%
Logarithmic differentiation	7	8%
Simplify the inverse-trig first, then differentiate	7	8%
Derivative of one function with respect to another	5	6%
Product and quotient rules	4	4%
Differentiating functional equations	2	2%
Trigonometric Simplification Toolkitfoundation	—	—

Parametric, Implicit & Higher-Order Derivatives20 PYQs · 22%

Concept	PYQs	Share
Parametric differentiation	6	7%
Implicit differentiation	5	6%
Logarithmic differentiation of implicit power relations	3	3%
Higher-order derivatives	3	3%
Second derivative of an inverse — the d²x/dy² identity	2	2%
Showing y satisfies a differential equation	1	1%

Differentiability — When the Derivative Exists18 PYQs · 20%

Concept	PYQs	Share
The left-hand = right-hand derivative test	7	8%
Modulus corners — and the x\|x\| trap	6	7%
Derivative at an awkward point via the limit definition	2	2%
Differentiable implies continuous (not the converse)	1	1%
Greatest-integer and step functions	1	1%
Finding parameters so f is differentiable	1	1%

Test yourself on Differentiation

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