NDA Maths · Teaching notes

Differentiation — NDA Mathematics

Differentiation is a high-volume NDA chapter — around 85 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.

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Subtopic notes

PYQ weightage by concept

20 concepts · 85 PYQs — where the marks actually sit, so you know what to drill first

Core Techniques — Standard Derivatives, Rules, Chain & Logarithmic49 PYQs · 58%

Concept	PYQs	Share
The derivative as a limit (first principles)	10	12%
The chain rule (composite functions)	9	11%
Logarithmic differentiation	7	8%
Simplify the inverse-trig first, then differentiate	7	8%
Standard derivatives to memorise	5	6%
Derivative of one function with respect to another	5	6%
Product and quotient rules	4	5%
Differentiating functional equations	2	2%

Parametric, Implicit & Higher-Order Derivatives20 PYQs · 24%

Concept	PYQs	Share
Parametric differentiation	6	7%
Implicit differentiation	5	6%
Logarithmic differentiation of implicit power relations	3	4%
Higher-order derivatives	3	4%
Second derivative of an inverse — the d²x/dy² identity	2	2%
Showing y satisfies a differential equation	1	1%

Differentiability — When the Derivative Exists16 PYQs · 19%

Concept	PYQs	Share
The left-hand = right-hand derivative test	7	8%
Modulus corners — and the x\|x\| trap	4	5%
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%

Formula & revision sheet

11 formulas · 1 reference tables · 6 gotchas across all subtopics — the exam-eve cheat-sheet

Core Techniques — Standard Derivatives, Rules, Chain & Logarithmic

Formulas (6)

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 $\frac{du}{dv} = \frac{du/dx}{dv/dx}$
Differentiating functional equations · Exponential functional equation $f(x+y)=f(x)f(y) \;\Rightarrow\; f'(x) = f'(0)\,f(x)$

Reference tables (1)

Standard derivatives to memorise11 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.
$\ln x$	$\dfrac{1}{x}$
$\log_a x$	$\dfrac{1}{x\ln a}$
$\sin^{-1} x$	$\dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$	$\dfrac{1}{1+x^2}$

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

Watch out for (2)

Degrees must be converted to radians first
→ Standard derivatives to memorise
Don't quotient-rule the raw inverse-trig
→ Simplify the inverse-trig first, then differentiate

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 (1)

Second derivatives don't invert like first derivatives
→ Second derivative of an inverse — the d²x/dy² identity

Differentiability — When the Derivative Exists

Formulas (1)

The left-hand = right-hand derivative test · One-sided derivative test $f'(c^-) = f'(c^+) \iff f \text{ differentiable at } c$

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