Differentiation on NET tests about fifteen derivative rules and the chain rule, under a clock of 54 seconds a question. Learn the tables below, know the five traps, and most of these MCQs take you under 30 seconds each.
Key facts
- Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
- Differentiation is unit 13 of the Class 11 PECTAA book; Further Differentiation is unit 2 of Class 12 (National Curriculum 2023)
- Our chapter map estimates about 9 MCQs from Differentiation and 6 from Further Differentiation. NUST does not publish chapter counts
Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map
What the two chapters cover
The Class 11 chapter gives you the rules for derivatives of algebraic, trigonometric, exponential, logarithmic and inverse trigonometric functions. It then uses them for rates of change, tangents, and maxima and minima.
The Class 12 chapter, Further Differentiation, adds the chain rule in full, implicit and parametric differentiation, higher-order derivatives, and the optimisation and approximation questions built on them.
On the paper the two blur together. A single MCQ can ask for a second derivative of a parametric curve, which needs both books. So prepare them as one topic.
The formulas you must know cold
d/dx x^n = n x^(n-1)
d/dx e^x = e^x d/dx a^x = a^x ln a
d/dx ln x = 1/x d/dx log_a x = 1/(x ln a)
d/dx sin x = cos x d/dx cos x = -sin x
d/dx tan x = sec^2 x d/dx cot x = -csc^2 x
d/dx sec x = sec x tan x d/dx csc x = -csc x cot x
d/dx sin^-1 x = 1/sqrt(1 - x^2)
d/dx cos^-1 x = -1/sqrt(1 - x^2)
d/dx tan^-1 x = 1/(1 + x^2)
The combining rules:
product (uv)' = u'v + uv'
quotient (u/v)' = (u'v - uv') / v^2
chain dy/dx = (dy/du)(du/dx)
parametric dy/dx = (dy/dt) / (dx/dt)
d2y/dx2 = [d/dt (dy/dx)] / (dx/dt)
w.r.t. g d f / d g = f'(x) / g'(x)
The nth derivative patterns that NET loves:
d^n/dx^n sin(ax + b) = a^n sin(ax + b + n*pi/2)
d^n/dx^n cos(ax + b) = a^n cos(ax + b + n*pi/2)
d^n/dx^n e^(ax) = a^n e^(ax)
For applications: the slope of the tangent at x = x0 is f'(x0) and the normal has slope -1/f'(x0). A function is increasing where f'(x) > 0. At a stationary point, f'' < 0 means a maximum and f'' > 0 means a minimum.
Worked MCQs
Five questions written in the style of our practice bank. We solved each by hand and checked it against the options.
1. Find d/dx cos x°.
The trap is the degree sign. Convert first: x° = (pi/180) x radians. By the chain rule:
d/dx cos(pi x / 180) = -(pi/180) sin(pi x / 180) = -(pi/180) sin x°
Answer: -(pi/180) sin x°. The option -sin x° is there to catch you.
2. Find d/dx x^x.
Neither the power rule nor the exponential rule applies, because both base and exponent vary. Take logs:
y = x^x
ln y = x ln x
(1/y) y' = ln x + 1
y' = x^x (1 + ln x)
3. If x = t^2 and y = t^3, find d2y/dx2.
dy/dx = 3t^2 / 2t = 3t/2
d2y/dx2 = d/dt(3t/2) / (dx/dt) = (3/2) / (2t) = 3/(4t)
With t = sqrt(x) (taking t > 0), the answer is 3/(4 sqrt(x)). The common wrong answer is 3/2, which comes from differentiating 3t/2 with respect to t and stopping there.
4. Where is f(x) = x^3 - 27x + 5 increasing?
f'(x) = 3x^2 - 27 > 0
x^2 > 9
x > 3 or x < -3, that is |x| > 3
5. Find the 777th derivative of cos x.
The derivatives of cos x cycle with period 4: -sin x, -cos x, sin x, cos x. Since 777 = 4 x 194 + 1, the 777th derivative equals the first: -sin x.
Common traps
- Degrees. Any angle written with a degree sign needs the
pi/180factor. The options often include the radian answer to catch you. - Stopping the chain early. In
d/dx cos(cos(cos x))there are three layers, so three factors:-sin(cos(cos x)) . sin(cos x) . sin x. Count the layers before you write. - Parametric second derivative. You must divide by
dx/dta second time. This is the step people skip. a^xvsx^a. One givesa^x ln a, the othera x^(a-1). Under time pressure students swap them.- "With respect to" another function.
d(sin x)/d(cos x)iscos x / (-sin x) = -cot x. Differentiate both with respect to x, then divide.
How to prepare
Start with the rule table above until you can write it from memory in two minutes. Then do mixed sets, not one rule at a time, because on the paper nobody tells you which rule a question needs. The maths formula sheet has every derivative on one page if you want something to pin above your desk.
Integration is the reverse of this chapter, and NET tests both. Once your derivatives are fast, move to the integration guide, where checking an answer means differentiating it. The limits guide covers the chapter that comes right before this one in the book.
FAQ
How many differentiation questions come in NET?
NUST does not publish chapter counts. Our chapter map estimates about 9 from Differentiation and 6 from Further Differentiation, which makes it one of the largest blocks in the Mathematics section.
Is further differentiation included in the NET syllabus?
Yes. NET is compiled from the HSSC textbooks for both years, and Further Differentiation is unit 2 of the Class 12 PECTAA book. See the NET syllabus for the full list.
Do I need to memorise inverse trigonometric derivatives for NET?
Yes. Questions on sin^-1 x and tan^-1 x appear often, usually inside a chain rule. Learn the three main ones and get the others from the sign pattern.
What is the fastest way to find a high-order derivative of sin or cos?
Divide the order by 4 and use the remainder. A remainder of 1 gives the first derivative, 2 the second, and so on. A remainder of 0 gives the function back.
