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Derivatives and Their Applications hero

Derivatives and Their Applications

~14 min read

In 30 seconds
  • What: Derivatives measure the rate of change of a function; applications include finding maxima, minima, tangents, normals, and rates of change in geometry and physics.
  • Why it matters: NDA papers from 2010 to 2025 carry roughly 10–14 questions per paper across both Derivatives and Application of Derivatives — one of the heaviest-weighted calculus topics.
  • Key fact: Chain rule, logarithmic differentiation, and the first-derivative test for maxima/minima together account for the majority of marks; master these first.

Differentiation sits at the heart of NDA Mathematics Paper II. Once you are comfortable finding derivatives using standard rules, the exam shifts to applying them — spotting increasing/decreasing intervals, locating maxima and minima, writing equations of tangents and normals, and solving real-world rate-of-change problems. This page walks you through every concept the NDA has actually tested, in the order the exam most often asks them.

What This Topic Covers

Derivatives and Their Applications is split across two chapters in NDA question banks: Derivatives (differentiation rules, higher-order derivatives, implicit and parametric differentiation) and Application of Derivatives (monotonicity, extrema, tangent/normal, and rate of change). Both chapters are tested together in the exam, and questions from each appear alongside each other in the same paper.

Core sub-topics the NDA tests

  • Standard derivatives — power, exponential, logarithmic, trigonometric, and inverse-trigonometric functions
  • Product rule, quotient rule, and chain rule — including compositions like \(\sin(\cos x)\) or \(e^{\sin^2 x}\)
  • Logarithmic differentiation — for functions of the form \([f(x)]^{g(x)}\), e.g., \((\cos x)^{\cos x}\) or \(x^x\)
  • Implicit differentiation — equations like \(x^m + y^m = 1\), \(x^n y^n = a^{m+n}\)
  • Parametric differentiation — curves given as \(x = f(t)\), \(y = g(t)\)
  • Higher-order derivatives — \(\frac{d^2y}{dx^2}\)
  • Increasing and decreasing functions — using the sign of \(f'(x)\)
  • Local maxima and minima — first-derivative test and second-derivative test
  • Absolute maxima/minima on a closed interval
  • Equations of tangent and normal to a curve
  • Rate of change — area, volume, radius problems
  • Derivative of one function with respect to another

Exam Pattern & Weightage

The table below is built from the PYQ data spanning 2010 to 2025. The combined question count for Derivatives + Application of Derivatives is consistently high — typically 10 to 14 questions per paper, split roughly 55:45 between the two sub-chapters.

Year Paper Derivatives Qs Application Qs Notable sub-types asked
2013 I & II 6 4 Implicit diff, modulus function derivative, log diff
2014 I & II 5 4 Parametric curves, tangent slope, box optimisation
2015 I & II 7 6 Chain rule, increasing/decreasing, maxima of parametric
2016 I & II 5 4 Higher-order, functional equations, f(8) min/max
2017 I & II 5 6 Log differentiation, increasing interval length, tangent angle
2018 I & II 4 5 Composite functions, flower-bed sector, rate of change
2019 I & II 4 4 Implicit diff x^y = e^(x-y), monotonicity intervals
2020 I 3 4 Derivative of tan⁻¹x w.r.t. cot⁻¹x, max of sin x − cos x
2021 I & II 5 5 GIF derivative, higher-order implicit, rate of change
2022 I & II 4 5 Log diff y=(x^x)^x, optimisation xy=4225, tangent angles
2023 I & II 4 3 |ln x| derivative, sin⁻¹x vs cos⁻¹x derivative
2024 I & II 4 4 Functional equations, f(x)=ln x/x monotonicity, log₉(x²+2x+11) min
2025 I & II 5 6 Parametric sec θ − cos θ, (x+y)^(p+q)=x^p y^q, triangle area optimisation
⚡ NDA Alert

Every paper from 2013 to 2025 contains at least one question on logarithmic differentiation (usually [f(x)]^g(x) type) and at least one maxima/minima problem. Never skip these two sub-types.

Core Concepts

Standard Derivative Rules

These are the building blocks. All NDA questions reduce to combinations of these rules. Memorise them cold.

Power Rule $$\frac{d}{dx}[x^n] = n \cdot x^{n-1}$$
Exponential & Log $$\frac{d}{dx}[e^x] = e^x \quad\Big|\quad \frac{d}{dx}[a^x] = a^x \ln a \quad\Big|\quad \frac{d}{dx}[\ln x] = \frac{1}{x} \quad\Big|\quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$$
Trigonometric Derivatives $$\frac{d}{dx}[\sin x] = \cos x \quad\Big|\quad \frac{d}{dx}[\cos x] = -\sin x \quad\Big|\quad \frac{d}{dx}[\tan x] = \sec^2 x$$ $$\frac{d}{dx}[\csc x] = -\csc x \cdot \cot x \quad\Big|\quad \frac{d}{dx}[\sec x] = \sec x \cdot \tan x \quad\Big|\quad \frac{d}{dx}[\cot x] = -\csc^2 x$$
Inverse Trigonometric Derivatives $$\frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}} \quad\Big|\quad \frac{d}{dx}[\cos^{-1} x] = -\frac{1}{\sqrt{1-x^2}}$$ $$\frac{d}{dx}[\tan^{-1} x] = \frac{1}{1+x^2} \quad\Big|\quad \frac{d}{dx}[\cot^{-1} x] = -\frac{1}{1+x^2}$$
Constant & Reciprocal Forms $$\frac{d}{dx}[c] = 0 \quad\Big|\quad \frac{d}{dx}[\sqrt{x}] = \frac{1}{2\sqrt{x}} \quad\Big|\quad \frac{d}{dx}\!\left[\frac{1}{x}\right] = -\frac{1}{x^2}$$

Memory hook for trig derivatives

  • Every "co-" function (\(\cos\), \(\cot\), \(\csc\), \(\cos^{-1}\), \(\cot^{-1}\)) picks up a negative sign on differentiation.
  • \(e^x\) is the only function whose derivative equals itself.
  • For \(a^x\), multiply by \(\ln a\) — a single year (e.g., 2017-II) of forgetting this loses one easy mark.

Product, Quotient, and Chain Rule

These three rules are the most-used in the Derivatives chapter. Expect chain rule inside every composite-function question.

Product Rule $$\frac{d}{dx}[u \cdot v] = u'v + uv'$$
Quotient Rule $$\frac{d}{dx}\!\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$$
Chain Rule $$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$ Example: $$\frac{d}{dx}[\sin(\cos x)] = \cos(\cos x) \cdot (-\sin x) = -\sin x \cdot \cos(\cos x)$$
⚡ NDA Alert

A 2013-II question asked for the derivative of \(\sin(\sin x)\). The answer is \(\cos(\sin x) \cdot \cos x\). Examiners test whether you apply chain rule inside a composite trig — a very common trap.

Logarithmic Differentiation

Use this whenever the function is of the form \([f(x)]^{g(x)}\) — both base and exponent are functions of \(x\). Take \(\ln\) of both sides first, then differentiate implicitly.

Log Differentiation — General Pattern $$y = [f(x)]^{g(x)} \implies \ln y = g(x) \cdot \ln[f(x)]$$ $$\frac{1}{y} \cdot \frac{dy}{dx} = g'(x) \cdot \ln[f(x)] + g(x) \cdot \frac{f'(x)}{f(x)}$$ Solve for \(\frac{dy}{dx}\) by multiplying both sides by \(y\).

A 2017-I question used \(y = (\cos x)^{\cos x}\). Taking \(\ln\) gives \(\ln y = \cos x \cdot \ln(\cos x)\). Differentiating: $$\frac{1}{y} \cdot y' = -\sin x \cdot \ln(\cos x) + \cos x \cdot \left(-\frac{\sin x}{\cos x}\right) = -\sin x\,[1 + \ln(\cos x)].$$ So \(y' = -y \sin x \,[1 + \ln(\cos x)]\) — exactly the form the answer choices presented.

⚡ NDA Alert

If the variable appears in both base and exponent (\(x^x\), \((\sin x)^x\), \(x^{\ln x}\)), the ordinary power rule fails — \(\frac{d}{dx}[x^x]\) is not \(x \cdot x^{x-1}\). Always take \(\ln\) of both sides first. Examiners include the wrong "power rule" answer as a distractor in nearly every paper.

Implicit and Parametric Differentiation

Implicit: differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\), then solve for \(\frac{dy}{dx}\). Parametric: if \(x = x(t)\) and \(y = y(t)\), then \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\).

Parametric Derivative $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \quad\Big|\quad \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\!\left(\frac{dy}{dx}\right)}{dx/dt}$$

A 2014-II set asked about \(x = a(\cos\theta + \theta \sin\theta)\) and \(y = a(\sin\theta - \theta \cos\theta)\). Differentiating: \(\frac{dx}{d\theta} = a\theta \cos\theta\) and \(\frac{dy}{d\theta} = a\theta \sin\theta\). So \(\frac{dy}{dx} = \tan\theta\). For \(\frac{d^2y}{dx^2}\), differentiate \(\frac{dy}{dx} = \tan\theta\) with respect to \(\theta\) and divide by \(\frac{dx}{d\theta} = a\theta \cos\theta\), giving \(\frac{d^2y}{dx^2} = \frac{\sec^2\theta}{a\theta \cos\theta}\).

Increasing and Decreasing Functions

A function \(f\) is increasing on an interval if \(f'(x) > 0\) there, and decreasing if \(f'(x) < 0\). To find these intervals: solve \(f'(x) = 0\) to get critical points, then check the sign of \(f'(x)\) in each sub-interval.

Monotonicity Test $$f'(x) > 0 \text{ on } (a, b) \implies f \text{ is strictly increasing on } (a, b)$$ $$f'(x) < 0 \text{ on } (a, b) \implies f \text{ is strictly decreasing on } (a, b)$$

A 2015-II question used \(f(x) = -2x^3 - 9x^2 - 12x + 1\). Differentiating: \(f'(x) = -6x^2 - 18x - 12 = -6(x^2 + 3x + 2) = -6(x + 1)(x + 2)\). Sign analysis gives \(f\) increasing on \((-2, -1)\) and decreasing on \((-\infty, -2) \cup (-1, \infty)\).

Maxima and Minima

At a local maximum, \(f'(x) = 0\) and \(f''(x) < 0\). At a local minimum, \(f'(x) = 0\) and \(f''(x) > 0\). When \(f''(x) = 0\) at the critical point, use the first-derivative test (sign change of \(f'\) across the point).

Second-Derivative Test $$f'(c) = 0 \text{ and } f''(c) < 0 \implies \text{local maximum at } x = c$$ $$f'(c) = 0 \text{ and } f''(c) > 0 \implies \text{local minimum at } x = c$$ $$f'(c) = 0 \text{ and } f''(c) = 0 \implies \text{test inconclusive; use first-derivative test}$$
⚡ NDA Alert

The 2011-I paper asked: at an extreme point of \(f(x)\), the tangent to the curve is parallel to the x-axis. This is because \(f'(x) = 0\) at a local extremum. Do not confuse this with a point of inflection, where \(f''(x) = 0\) but \(f'\) does not change sign.

Tangent and Normal to a Curve

The slope of the tangent to \(y = f(x)\) at \((x_0, y_0)\) is \(m = f'(x_0)\). The equation of the tangent is: \(y - y_0 = m(x - x_0)\). The normal is perpendicular to the tangent, so its slope is \(-\frac{1}{m}\).

Tangent and Normal Equations $$\text{Tangent: } y - y_0 = f'(x_0)(x - x_0)$$ $$\text{Normal: } y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$$
Special Geometric Conditions Tangent parallel to x-axis → $$\frac{dy}{dx} = 0$$
Tangent perpendicular to x-axis → $$\frac{dx}{dy} = 0$$ (i.e., $$\frac{dy}{dx}$$ is infinite)
Two curves cut orthogonally → $$m_1 \cdot m_2 = -1$$ at the point of intersection
⚡ NDA Alert

Tangent and normal are perpendicular: slope of normal = \(-1/\)(slope of tangent). A common 2-mark trap is mixing the two — if asked for the normal, never write \(y - y_0 = f'(x_0)(x - x_0)\). Also remember: the shortest distance between two non-intersecting curves lies along their common normal.

Rate of Change

If a quantity \(Q\) depends on time \(t\), its rate of change is \(\frac{dQ}{dt}\). For linked quantities (e.g., radius and area of a circle), use the chain rule: \(\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}\).

Circle Rate-of-Change $$A = \pi r^2 \implies \frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}$$ $$\text{Circumference } C = 2\pi r \implies \frac{dC}{dt} = 2\pi \cdot \frac{dr}{dt}$$

A 2020-I question: radius increasing at 0.7 cm/s. Rate of increase of circumference \(= 2\pi \times 0.7 = 1.4\pi \approx 4.4\) cm/s.

See also: Limits, Continuity & Differentiability — that topic feeds directly into this one. For what comes next in the calculus sequence, visit Indefinite and Definite Integration.

Worked Examples

These seven examples come from NDA PYQ papers and standard exam patterns. Work through each step before reading the next — that is the fastest way to build exam speed.

Example 1 — Derivative of sec²x with respect to tan²x (2013-I)

Find the derivative of \(u = \sec^2 x\) with respect to \(v = \tan^2 x\).

  • Write $$\frac{du}{dx} = 2 \sec x \cdot \sec x \tan x = 2 \sec^2 x \tan x.$$
  • Write $$\frac{dv}{dx} = 2 \tan x \cdot \sec^2 x.$$
  • $$\frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{2 \sec^2 x \tan x}{2 \tan x \sec^2 x} = 1.$$
  • Answer: \(1\). The derivative of \(\sec^2 x\) with respect to \(\tan^2 x\) is always \(1\), because \(\sec^2 x = 1 + \tan^2 x\), so \(u = 1 + v\) and \(\frac{du}{dv} = 1\).

Example 2 — Derivative of sin(sin x) (2013-II)

What is the derivative of \(\sin(\sin x)\)?

  • Let \(y = \sin(\sin x)\). Apply the chain rule: the outer function is \(\sin(\cdot)\) and the inner function is \(\sin x\).
  • $$\frac{dy}{dx} = \cos(\sin x) \cdot \frac{d}{dx}(\sin x) = \cos(\sin x) \cdot \cos x.$$
  • Answer: \(\cos(\sin x) \cdot \cos x\). This is the chain rule applied twice — first differentiate \(\sin(\cdot)\) to get \(\cos(\cdot)\), then multiply by the derivative of the inner function.

Example 3 — Implicit differentiation: x^y = e^(x−y) (2019-II)

Find \(\frac{dy}{dx}\) if \(x^y = e^{x-y}\) and evaluate at \(x = 1\).

  • Take \(\ln\) of both sides: \(y \ln x = (x - y) \cdot 1\), so \(y \ln x = x - y\).
  • Rearrange: \(y \ln x + y = x \implies y(1 + \ln x) = x \implies y = \dfrac{x}{1 + \ln x}\).
  • At \(x = 1\): \(y = \dfrac{1}{1 + 0} = 1\). So the point is \((1, 1)\).
  • Differentiate \(y(1 + \ln x) = x\) implicitly: $$y'(1 + \ln x) + y \cdot \frac{1}{x} = 1.$$
  • At \((1, 1)\): \(y'(1 + 0) + 1 \cdot 1 = 1 \implies y' + 1 = 1 \implies y' = 0\).
  • Answer: \(\frac{dy}{dx}\) at \(x = 1\) is \(0\).

Example 4 — Maxima/minima: largest value on [−2, 2] (2011-I)

Find the largest value of \(2x^3 - 3x^2 - 12x + 5\) for \(-2 \le x \le 2\).

  • Let \(f(x) = 2x^3 - 3x^2 - 12x + 5\). Find $$f'(x) = 6x^2 - 6x - 12 = 6(x^2 - x - 2) = 6(x - 2)(x + 1).$$
  • Critical points inside \([-2, 2]\): \(x = -1\) and \(x = 2\).
  • Evaluate \(f\) at all endpoints and critical points: \(f(-2) = -16 - 12 + 24 + 5 = 1\); \(f(-1) = -2 - 3 + 12 + 5 = 12\); \(f(2) = 16 - 12 - 24 + 5 = -15\).
  • The largest value is \(12\), occurring at \(x = -1\).

Example 5 — Chain rule on a composite trig function

Differentiate \(y = \sin(x^2 + 5)\).

  • Identify the outer function (\(\sin\)) and the inner function (\(x^2 + 5\)).
  • Differentiate the outer, keeping the inner intact: $$\cos(x^2 + 5).$$
  • Multiply by the derivative of the inner: $$\frac{d}{dx}(x^2 + 5) = 2x.$$
  • Answer: $$\frac{dy}{dx} = 2x \cos(x^2 + 5).$$ The chain rule is the single most-tested rule in this chapter — every composite function reduces to this two-step pattern.

Example 6 — Second-derivative test for local minimum

Find the local minimum value of \(f(x) = x^3 - 3x + 2\).

  • First derivative: \(f'(x) = 3x^2 - 3\). Set \(f'(x) = 0 \implies x^2 = 1 \implies\) critical points \(x = 1, x = -1\).
  • Second derivative: \(f''(x) = 6x\).
  • At \(x = 1\): \(f''(1) = 6 > 0 \implies\) local minimum. At \(x = -1\): \(f''(-1) = -6 < 0 \implies\) local maximum.
  • Local minimum value \(= f(1) = 1 - 3 + 2 = 0\). Answer: \(0\).

Example 7 — Rate of change: area of circle (2012-II)

The radius of a circle increases uniformly at 3 cm/s. What is the rate of increase in area when the radius is 10 cm?

  • Area \(A = \pi r^2\). Differentiate: $$\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}.$$
  • Given: \(\frac{dr}{dt} = 3\) cm/s, \(r = 10\) cm.
  • $$\frac{dA}{dt} = 2\pi \times 10 \times 3 = 60\pi \text{ cm}^2/\text{s}.$$
  • Answer: \(60\pi\) cm²/s. Always write out the chain rule step — many candidates lose marks by substituting numbers before differentiating.

Common Question Patterns

Recognising the question type saves you 30–60 seconds per question. Here are the six patterns the NDA has repeated most often across the 2010–2025 papers.

Pattern 1 — Derivative of one function w.r.t. another

You will be asked: "What is the derivative of \(f(x)\) with respect to \(g(x)\)?" Use the chain rule: \(\frac{df}{dg} = \frac{df/dx}{dg/dx}\). The 2013-I paper had \(\sec^2 x\) vs \(\tan^2 x\); the 2015-I paper had a \(\tan^{-1}\) expression vs \(\tan^{-1} x\); the 2019-II paper had \(2^{\sin x}\) vs \(\sin x\).

Pattern 2 — Logarithmic differentiation for [f(x)]^g(x)

Whenever the base and exponent both contain \(x\), take log first. Common NDA examples: \(y = x^x\) (at \(x = 1\)), \(y = (\cos x)^{\cos x}\) (2017-I), \(y = (x^x)^x = x^{x^2}\) (2022-I). After taking \(\ln\), the question reduces to a product-rule or chain-rule calculation.

Pattern 3 — Modulus function or greatest integer function derivative

Check whether the derivative exists at the given point. For \(|x|\) at \(x = 0\), left-hand derivative is \(-1\) and right-hand derivative is \(+1\), so the derivative does not exist (2013-I). For the greatest integer function, the derivative is \(0\) wherever the function is constant — which is everywhere except the integers (2022-I).

Pattern 4 — Increasing/decreasing interval of a polynomial or composite

Factorise \(f'(x)\) and use a sign chart. The NDA also asks about standard functions: \(\ln x\) is increasing on \((0, \infty)\); \(e^x\) is increasing everywhere; \(\tan x\) is increasing on each interval $$\left(-\frac{\pi}{2} + n\pi,\ \frac{\pi}{2} + n\pi\right)$$ but is not monotone on all of \(\mathbb{R}\) (2020-I asked about this directly).

Pattern 5 — Maxima/minima of a⋅sin x + b⋅cos x type

The maximum value of \(a \sin x + b \cos x\) is \(\sqrt{a^2 + b^2}\). The 2017-II match-list question had four such functions: \(\sin x + \cos x\) has max \(\sqrt{2}\); \(3 \sin x + 4 \cos x\) has max \(5\); \(2 \sin x + \cos x\) has max \(\sqrt{5}\); \(\sin x + \sqrt{3} \cos x\) has max \(2\).

Max of a·sin x + b·cos x $$\text{Maximum value} = \sqrt{a^2 + b^2} \quad\Big|\quad \text{Minimum value} = -\sqrt{a^2 + b^2}$$

Pattern 6 — Geometry/physics optimisation

Describe a constraint (fixed perimeter, fixed surface area, given length of wire), express the quantity to optimise as a function of one variable, differentiate, and set equal to zero. Classic NDA examples: inscribed cylinder in sphere (2014-II), rectangular box from sheet (2014-II), cylindrical jar without lid (2017-II), and flower-bed sector of wire length 40 m (2018-II, answer: radius = 10 m).

How NDA Tests This Topic

About 30% of questions are pure differentiation (apply a rule to a given function). Another 30% are increasing/decreasing or maxima/minima. The remaining 40% are applied problems: rate of change, tangent/normal, or parametric curves. There is almost always a set of 2–3 linked questions on a single function or curve — read all parts before answering any of them.

Exam Shortcuts (Pro-Tips)

Calculus questions in the NDA are designed so that mechanical chain-rule grinding eats up your time bank. The seven shortcuts below collapse 3–5 minutes of working into 10–20 seconds each — every one has a direct match in a recent NDA paper.

Shortcut 1 — Infinite Nested Square Root

NDA loves infinite square-root series. Never square both sides; the direct formula is faster.

Infinite Nested Root If $$y = \sqrt{f(x) + \sqrt{f(x) + \sqrt{f(x) + \dots \infty}}}$$, then $$\frac{dy}{dx} = \frac{f'(x)}{2y - 1}$$

Example: if $$y = \sqrt{\sin x + \sqrt{\sin x + \dots}}$$, the answer is instantly $$\frac{\cos x}{2y - 1}$$.

Shortcut 2 — Infinite Power Tower

For an infinite power-tower of the same function, use the closed form directly:

Infinite Power Series If $$y = f(x)^{f(x)^{f(x)^{\dots \infty}}}$$, then $$\frac{dy}{dx} = \frac{y^2 \, f'(x)}{f(x)\,[1 - y \ln f(x)]}$$

Shortcut 3 — Inverse Trig Substitution Trick

For ugly inverse-trig expressions like $$\tan^{-1}\!\left(\frac{2x}{1 - x^2}\right)$$, do not chain-rule. Substitute $$x = \tan\theta$$ — the expression collapses to a double/half-angle identity.

Common Inverse-Trig Reductions $$\tan^{-1}\!\left(\frac{2x}{1-x^2}\right) = 2\tan^{-1}x$$  |  $$\sin^{-1}(2x\sqrt{1-x^2}) = 2\sin^{-1}x$$
So $$\frac{d}{dx}\tan^{-1}\!\left(\frac{2x}{1-x^2}\right) = \frac{2}{1+x^2}$$ — instant.

Shortcut 4 — Algebraic Optimisation Without Calculus

If a question asks to divide a number K into two parts to maximise their product, never differentiate. The optimum split is symmetric.

Product-Maximisation Rule Maximise $$xy$$ with $$x + y = K$$ → $$x = y = K/2$$
Maximise $$x^p y^q$$ with $$x + y = K$$ → $$x = K\!\left(\frac{p}{p+q}\right)$$, $$y = K\!\left(\frac{q}{p+q}\right)$$

Shortcut 5 — Standard Geometric Maxima (Memorise)

UPSC repeats these inscribed-figure results. Knowing the answer skips a full page of differentiation.

Direct results worth memorising

  • Rectangle of maximum area inscribed in a circle → always a square.
  • Cylinder of maximum volume inscribed in a sphere of radius R → height = $$\frac{2R}{\sqrt{3}}$$.
  • Cone of maximum volume inscribed in a sphere of radius R → height = $$\frac{4R}{3}$$.
  • Maximum of $$a \sin x + b \cos x$$ = $$\sqrt{a^2 + b^2}$$; minimum = $$-\sqrt{a^2 + b^2}$$.

Shortcut 6 — Point of Inflection in One Step

If a question asks for the point of inflection (curvature change), just solve f''(x) = 0. No need for the first-derivative test or sign chart — that's only for max/min.

Inflection vs Extremum $$\text{Local max/min} \implies f'(x) = 0 \text{ (then test sign of } f''(x)\text{)}$$ $$\text{Point of inflection} \implies f''(x) = 0$$

Shortcut 7 — Parametric Second Derivative (Avoid the Trap)

For parametric x = x(t), y = y(t), the second derivative is not (d²y/dt²)/(d²x/dt²). That is the single most common error in this chapter.

Correct Parametric d²y/dx² $$\frac{d^2y}{dx^2} = \frac{d}{dt}\!\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx}$$

Differentiate the first derivative dy/dx with respect to t, then multiply by 1/(dx/dt). Skipping the dt/dx factor is the trap NDA setters plant in match-the-following questions.

Preparation Strategy

Use this four-week plan if you are starting from scratch on this topic.

Week 1 — Rules and standard derivatives

Drill the formula card for standard derivatives until you recall all 12 without hesitation. Then practice product, quotient, and chain rule on 20 questions per day. Focus on composite trig functions — \(\sin(\cos x)\), \(\cos(\sin x)\), \(e^{\sin x}\) — since these appear almost every year. Link to: Inverse Trigonometric Functions — the derivatives of \(\sin^{-1} x\) and \(\tan^{-1} x\) are tested very frequently in this chapter.

Week 2 — Logarithmic, implicit, and parametric

Work through every \([f(x)]^{g(x)}\) question from the PYQ file. There are at least eight distinct examples from 2013 to 2025. For implicit differentiation, practice \(x^m y^n = a^{m+n}\) type equations. For parametric, always write \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) clearly before computing the ratio.

Week 3 — Applications: monotonicity and extrema

Take any polynomial cubic, find critical points, and classify them — do this for 15 different functions. Then move to standard function behaviour: when is \(f(x) = x + \frac{1}{x}\) increasing? (Answer: on \((1, \infty)\), since \(f'(x) = 1 - \frac{1}{x^2} = 0\) at \(x = \pm 1\), and \(f'' > 0\) at \(x = 1\).) Practice the \(\sqrt{a^2 + b^2}\) formula for trigonometric maximum values. Link to: Differential Equations — rate-of-change language appears in both topics.

Week 4 — Tangent/normal and full mock practice

Write the equation of tangent and normal for five different curves. Then attempt the last three years of NDA papers (2023, 2024, 2025) under timed conditions — 90 seconds per question. Review every wrong answer by tracing back to which rule you applied incorrectly. Take a mock test to benchmark your accuracy before the exam. For a broader view of how this topic fits into the NDA Maths syllabus, visit the NDA Maths subject page.

Time allocation in the exam

Target 90 seconds per differentiation question and 2 minutes per application question. Multi-part question sets (the 2–3 linked questions) are faster once you resolve the curve properties in the first part — do not re-derive for each sub-part. If you are stuck on a logarithmic differentiation question for more than 2 minutes, skip and return — partial marks are not given in NDA, so a wrong guess and a skip have the same score outcome. A timed mock helps you calibrate this instinct.

Test yourself on Derivatives before the exam

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Frequently Asked Questions

How many questions from Derivatives appear in the NDA exam?

Based on the PYQ data from 2010 to 2025, the combined Derivatives + Application of Derivatives chapter typically contributes 10 to 14 questions per paper. Some years (2015, 2017, 2025) have gone as high as 13 questions across both papers combined. It is one of the highest-weighted individual topics in the NDA Maths section.

Is logarithmic differentiation always needed for y = x^x type questions?

Yes. Whenever both the base and exponent are functions of \(x\) — such as \(x^x\), \((\cos x)^{\cos x}\), or \((x^x)^x\) — you must take the natural log of both sides first. The NDA has tested this pattern repeatedly (2017-I, 2022-I, 2022-II). After taking \(\ln\), the problem reduces to a standard implicit or product-rule calculation.

Does the derivative of |x| exist at x = 0?

No. The left-hand derivative at \(x = 0\) is \(-1\) and the right-hand derivative is \(+1\). Since these are unequal, the derivative does not exist at \(x = 0\). This was directly asked in the 2013-I paper. More generally, the derivative of \(|x - a|\) does not exist at \(x = a\) — a point that recurs in modulus-function questions.

What is the maximum value of a·sin x + b·cos x?

The maximum value is \(\sqrt{a^2 + b^2}\) and the minimum value is \(-\sqrt{a^2 + b^2}\). For example, \(3 \sin x + 4 \cos x\) has maximum \(\sqrt{9 + 16} = \sqrt{25} = 5\). This formula appeared as a match-list question in the 2017-II paper with four different functions. Knowing this formula eliminates the need for calculus in most such questions.

How do I find the slope of a parametric curve at a given point?

Compute \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) separately, then divide: slope \(= \frac{dy}{dx} = \frac{dy/dt}{dx/dt}\). Substitute the given parameter value (the value of \(t\) or \(\theta\)) after computing the ratio. The 2014-II paper set used \(x = a(\cos\theta + \theta \sin\theta)\) and \(y = a(\sin\theta - \theta \cos\theta)\), giving \(\frac{dy}{dx} = \tan\theta\). For \(\frac{d^2y}{dx^2}\), differentiate \(\frac{dy}{dx}\) with respect to the parameter and divide again by \(\frac{dx}{dt}\).

What is the difference between a local maximum and an absolute maximum?

A local maximum is the highest value of \(f\) in a small neighbourhood around a point. An absolute maximum is the highest value of \(f\) on the entire domain or a specified closed interval. On a closed interval \([a, b]\), always check critical points inside the interval and both endpoints — the NDA tested this distinction explicitly in the 2011-I and 2013-II papers.

How is the derivative used to find the rate of change of volume or surface area?

Express the quantity (volume, surface area, circumference) as a function of a single variable (usually radius or side length), then differentiate with respect to time using the chain rule. For a sphere: \(V = \frac{4}{3}\pi r^3\), so \(\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt}\). The 2010-I and 2012-II papers had questions about spherical soap bubbles and circles using exactly this approach.