Differential Calculus — NDA Mathematics

NDA calculus is Class 11–12 CBSE calculus with the proofs stripped out. You are never asked to derive a theorem from scratch in 75 seconds — you are asked to recognise which standard limit, which derivative rule, or which application a question is built on, then execute it cleanly. The syllabus is short: limits, continuity, derivatives (first principles and standard forms), differentiation rules, and three applications. Master that short list and this chapter converts almost mechanically into marks.

1. What NDA actually asks

13% weight ⇒ roughly 15–16 of the 120 Mathematics questions, at 2.5 marks each with a −0.8333 penalty for a wrong guess. That makes Differential Calculus the fourth-heaviest topic after Algebra (20%), Trigonometry (18%), and Analytical Geometry (15%) — and among the most execution-friendly, because every question reduces to one of a handful of fixed procedures:

1. Standard limits — evaluating ${\lim }_{x\to 0}\frac{\sin kx}{x}$-type forms, algebraic $\frac{0}{0}$ limits via factoring, and limits at infinity.
2. Continuity — finding an unknown constant that makes a piecewise function continuous at a point.
3. Derivative from first principles — the ${\lim }_{h\to 0}$ definition applied to a simple function.
4. Standard derivatives — the fixed list for powers, trig, exponential, and log functions.
5. Differentiation rules — product, quotient, and chain rule, often chained two deep.
6. Second-order derivatives — differentiate twice, usually followed by an evaluation.
7. Applications — slope/equation of a tangent or normal, rate-of-change word problems, and basic maxima–minima (first- or second-derivative test).

There is no integration, no L'Hôpital's rule, and no rigorous $ϵ$–$\delta$ proof anywhere on the syllabus — NDA tests fluency with a toolkit, not analysis.

2. Functions, limits & continuity

A real-valued function $f:D\to \mathbb{R}$ assigns one real output to each input in its domain $D$. NDA rarely tests domain theory in isolation, but it underlies every limit and derivative question — know where a function is undefined (denominators zero, logs of non-positive numbers, even roots of negatives) before you differentiate it.

The limit ${\lim }_{x\to a}f(x)=L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ approaches $a$ — from both sides. Two limit laws you'll use constantly: limits of sums/products/quotients split termwise (provided the denominator's limit isn't zero), and for $\frac{0}{0}$ forms, factor and cancel before substituting.

Standard limits to know cold (all as $x\to 0$ unless stated):

${\lim }_{x\to 0}\frac{\sin x}{x}=1{\lim }_{x\to 0}\frac{\tan x}{x}=1{\lim }_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}$

${\lim }_{x\to 0}\frac{e^x-1}{x}=1{\lim }_{x\to 0}\frac{\ln (1+x)}{x}=1{\lim }_{x\to 0}\frac{a^x-1}{x}=\ln a$

${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}{\lim }_{x\to \infty }{\left(1+\frac{1}{x}\right)}^x=e$

Every "$\sin kx/x$" question is the first identity in disguise: multiply and divide by $k$ so the argument of sine matches the denominator.

Continuity at $x=a$ requires all three to hold and agree:

${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=f(a)$

NDA's favourite continuity question gives a piecewise function — a rational expression for $x\ne a$ and a constant $k$ at $x=a$ — and asks for the $k$ that removes the discontinuity. Simplify the $\frac{0}{0}$ form algebraically, take the limit, and set $k$ equal to it.

Continuous ≠ differentiable. $f(x)=|x|$ is continuous everywhere but has no derivative at $x=0$ (the left- and right-hand slopes disagree: $-1$ vs $+1$). Differentiability is the stronger condition; NDA occasionally tests this exact fact conceptually.

3. The derivative — first principles and the standard list

The derivative of $f$ at $x$ is defined as the limit of the average rate of change over a shrinking interval:

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Worked derivation (know this pattern): for $f(x)=\sin x$, $f^{\prime }(x)={\lim }_{h\to 0}\frac{\sin (x+h)-\sin x}{h}={\lim }_{h\to 0}\frac{2\cos \left(x+\frac{h}{2}\right)\sin \frac{h}{2}}{h}=\cos x\cdot {\lim }_{h\to 0}\frac{\sin (h/2)}{h/2}=\cos x$

using the sum-to-product identity and the standard limit $\lim \sin \theta /\theta =1$. NDA typically asks for one such derivation directly (usually $x^2$, $\sqrt{x}$, or $1/x$) rather than $\sin x$, but the method — expand, simplify the $h$ in the numerator, cancel, then take the limit — is identical for every function on the syllabus.

The standard derivative table (memorise; every differentiation question builds on this):

4. Differentiation rules & second-order derivatives

Real questions rarely differentiate a bare standard function — they combine two or three functions, which is where the three rules come in.

Sum/difference: $(u\pm v{)}^{\prime }=u^{\prime }\pm v^{\prime }$ — split and differentiate termwise.

Product rule: for $y=u(x)v(x)$, $\frac{dy}{dx}=u^{\prime }v+u{v}^{\prime }$

Quotient rule: for $y=\frac{u(x)}{v(x)}$, $\frac{dy}{dx}=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$

Chain rule: for a composite $y=f(g(x))$, treat $u=g(x)$ as the inner function: $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$

In practice: differentiate the outer function, keep the inner function unchanged inside it, then multiply by the derivative of the inner function. For $y=\sin (3x^2+1)$: outer derivative is $\cos (3x^2+1)$, inner derivative is $6x$, so $y^{\prime }=6x\cos (3x^2+1)$.

Second-order derivatives. Differentiate twice: $y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$. NDA uses this both as a standalone "differentiate twice and evaluate" question and inside the second-derivative test for maxima/minima (Section 6).

5. Applications — tangents, normals & rate of change

Tangent and normal at a point $(x_1,y_1)$ on $y=f(x)$: the derivative evaluated at $x_1$ is the slope of the tangent, $m=f^{\prime }(x_1)$.

$\text{Tangent: }y-y_1=m(x-x_1)\text{Normal: }y-y_1=-\frac{1}{m}(x-x_1)$

The normal is perpendicular to the tangent, so its slope is the negative reciprocal — never the same sign or the same value as $m$. If $m=0$ (horizontal tangent), the normal is the vertical line $x=x_1$.

Rate of change. $\frac{dy}{dx}$ is the instantaneous rate at which $y$ changes with $x$. When a quantity changes with time, chain-rule it: if $y$ depends on $x$ and $x$ depends on $t$, $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$

This is how "the radius of a circle grows at 3 cm/s — how fast does the area grow?" becomes $\frac{dA}{dt}=\frac{dA}{dr}\cdot \frac{dr}{dt}=2\pi r\cdot \frac{dr}{dt}$: differentiate the geometric formula with respect to the variable that's actually changing, then plug in the given rate and the instant's value.

Increasing/decreasing: $f$ is increasing on an interval where $f^{\prime }(x)>0$ and decreasing where $f^{\prime }(x)<0$ throughout that interval.

6. Applications — maxima and minima

A critical point of $f$ is where $f^{\prime }(x)=0$ (or $f^{\prime }$ fails to exist). NDA's maxima–minima questions are almost always: differentiate, set $f^{\prime }(x)=0$, solve for critical points, then classify each one.

Second-derivative test (the faster of the two, and NDA's default expectation):

• $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)<0\Rightarrow$ local maximum at $c$.
• $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)>0\Rightarrow$ local minimum at $c$.
• $f^{\prime \prime }(c)=0\Rightarrow$ inconclusive — fall back to the first-derivative test.

First-derivative test (use when $f^{\prime \prime }(c)=0$ or as a cross-check): if $f^{\prime }$ changes sign from $+$ to $-$ at $c$, it's a local max; from $-$ to $+$, a local min; no sign change means neither.

Once you've classified a critical point, plug it back into the original function $f$ — not $f^{\prime }$ — to get the actual maximum or minimum value. This last step is the single most common place marks are lost.

8. Common traps & training protocol

• Forgetting the chain-rule multiplier. Differentiating $\sin (2x^2+1)$ as $\cos (2x^2+1)$ alone — the inner derivative $4x$ is not optional.
• Sign error on $\cos x$. $\frac{d}{dx}\cos x=-\sin x$, not $+\sin x$; this single sign error cascades through every product/quotient question involving cosine.
• Quotient-rule order. It's $\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$, never $u{v}^{\prime }-u^{\prime }v$ — the numerator order matters because subtraction isn't commutative.
• Confusing tangent and normal slopes. The normal's slope is $-1/m$, not $m$ and not $1/m$ — a plain reciprocal without the sign flip is one of the most-picked wrong options.
• Second-derivative sign confusion. $f^{\prime \prime }(c)<0$ is a maximum (the curve bends downward, like a frown), $f^{\prime \prime }(c)>0$ is a minimum — many aspirants swap these under time pressure.
• Reporting the critical point instead of the extreme value. The question asks for the maximum/minimum value of $f$, which means substituting the critical point back into $f(x)$, not stopping at $x=c$.
• Dropping the $\pm$/direction in rate-of-change problems. A quantity that is decreasing has a negative rate; plug the given rate in with its correct sign before differentiating the relationship.

Training protocol: write out the standard-limit table and the standard-derivative table from memory daily until both are reflexive — most calculus marks are lost to a forgotten $-3k$-style constant, not to conceptual gaps. Then drill the three differentiation rules on composite functions (two rules stacked in one question is the norm, not the exception), and finish every maxima–minima problem by evaluating $f$ at the classified point, never at $f^{\prime }$ or $f^{\prime \prime }$.