8.1 Differentiation

Why Differentiation Appears on the PAF CAE Academic Paper

Mathematics is commonly weighted around 30% of the academic portion for Pakistan Air Force College of Aeronautical Engineering (CAE) and related engineering-branch initial tests. Coaching and candidate reports often describe a computer-based maths paper of roughly 50 MCQs in about 25 minutes (confirm your cycle’s slip on joinpaf.gov.pk). Under that pace you need instant recall of derivative rules, not lengthy ε–δ arguments.

This section covers the FSc Pre-Engineering toolkit: limits as the idea behind the derivative, the power, product, quotient, and chain rules, tangent lines, and rates of change. Integration and probability follow in later sections of this chapter.

Limits: The Idea Behind the Derivative

The average rate of change of y = f(x) from x = a to x = a + h is

[\frac{f(a+h) - f(a)}{h}.]

The derivative at x = a is the limit of that quotient as h approaches 0 (when the limit exists):

[f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.]

Geometrically, f′(a) is the slope of the tangent to the curve at the point (a, f(a)). Physically, if s(t) is position, s′(t) is instantaneous velocity.

Worked limit definition. Let f(x) = x². Then

[f'(3) = \lim_{h \to 0} \frac{(3+h)^2 - 9}{h} = \lim_{h \to 0} \frac{9 + 6h + h^2 - 9}{h} = \lim_{h \to 0}(6 + h) = 6.]

So the tangent slope at x = 3 is 6. For exam speed you will almost always use rules below instead of expanding the difference quotient—but knowing the definition explains why those rules exist.

One-sided / continuity note (MCQ trap). Differentiability at a requires the two-sided limit of the difference quotient to exist, which forces continuity at a. Continuity alone does not guarantee differentiability (classic corner: f(x) = |x| at 0).

Standard Derivative Rules (Memorize Cold)

Common FSc forms:

• d/dx[sin x] = cos x, d/dx[cos x] = −sin x, d/dx[tan x] = sec² x
• d/dx[eˣ] = eˣ, d/dx[ln x] = 1/x (x > 0)
• d/dx[aˣ] = aˣ ln a (when a > 0, a ≠ 1)

Worked power rule. Differentiate y = 5x⁴ − 3x² + 7. Then y′ = 20x³ − 6x. At x = 1, y′(1) = 14.

Worked product rule. Let u = x², v = sin x. Then y = x² sin x gives

y′ = 2x sin x + x² cos x.

At x = π/2: y′(π/2) = 2·(π/2)·1 + (π/2)²·0 = π.

Worked quotient rule. Let y = (3x + 1)/(x − 2). Then u = 3x + 1, v = x − 2,

u′ = 3, v′ = 1, so

y′ = [3(x − 2) − (3x + 1)(1)] / (x − 2)² = (3x − 6 − 3x − 1)/(x − 2)² = −7/(x − 2)².

At x = 0, y′(0) = −7/4.

Worked chain rule. Differentiate y = (2x³ − 5)⁵. Outer power 5, inner g = 2x³ − 5:

y′ = 5(2x³ − 5)⁴ · 6x² = 30x²(2x³ − 5)⁴.

At x = 1: y′(1) = 30(2 − 5)⁴ = 30 · 81 = 2430.

Nested chain (exam favorite). y = sin(3x²). Then y′ = cos(3x²) · 6x = 6x cos(3x²).

Tangents

If f is differentiable at a, the tangent line at (a, f(a)) is

[y - f(a) = f'(a),(x - a).]

Worked tangent. For f(x) = x³ − 2x at a = 2: f(2) = 8 − 4 = 4, f′(x) = 3x² − 2, f′(2) = 12 − 2 = 10. Tangent:

y − 4 = 10(x − 2) ⇒ y = 10x − 16.

Normal line (occasional): slope −1/f′(a) when f′(a) ≠ 0. At the point above, normal slope = −1/10, so y − 4 = (−1/10)(x − 2).

Rates of Change and Related Rates

If a quantity Q depends on time t, dQ/dt is its instantaneous rate. When two variables are linked by an equation, differentiate both sides with respect to t (chain rule), then substitute known values.

Worked related rates. A circular oil slick has radius r(t) with area A = πr². If r increases at 0.2 m/s when r = 5 m, find dA/dt.

Differentiating: dA/dt = 2πr · dr/dt = 2π·5·0.2 = 2π m²/s ≈ 6.28 m²/s.

Ladder / sliding object pattern. A 13 m ladder leans against a wall; base moves away at 0.5 m/s when base is 5 m from the wall. Then x² + y² = 169. Differentiate: 2x dx/dt + 2y dy/dt = 0 ⇒ dy/dt = −(x/y) dx/dt. When x = 5, y = 12, so dy/dt = −(5/12)(0.5) = −5/24 m/s (negative = descending).

On the Exam

Expect stems such as “f′(2) = ?”, “slope of the tangent at …”, “differentiate (3x − 1)⁴”, and short related-rate numbers. Skip multi-step geometry rates if a pure rule question is clearer—return only if time remains. Always simplify before plugging x = a when the algebra is messy; many wrong options are “forgot the chain factor” or “used product instead of quotient.”