Skip to main content
Mathematics 3% exam weight

Differentiation

Part of the CUET UG study roadmap. Mathematics topic math-019 of Mathematics.

→ See in Roadmap Build 1-Month Plan

Differentiation

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision.

Differentiation — Key Facts for CUET • Power Rule: d/dx(xⁿ) = nxⁿ⁻¹, valid for any real exponent n. For n = 0, derivative of constant 1 is 0. • Most tested concept: Chain Rule for composite functions — d/dx[f(g(x))] = f’(g(x)) · g’(x). Always multiply by the derivative of the inner function. • Common mistake: Forgetting to apply the product rule (d/dx[uv] = u’v + uv’) and quotient rule (d/dx[u/v] = (u’v − uv’)/v²) when differentiating products or quotients. • Key technique: Logarithmic differentiation — take ln of both sides for functions of the form y = [f(x)]^{g(x)} to simplify differentiation. • Important exception: Derivative of ln|f(x)| is f’(x)/f(x). The absolute value inside ln ensures the argument is positive, but the derivative formula ignores the sign because ln is defined only for positive arguments, so we write d/dx[ln|f(x)|] = f’(x)/f(x). • Most frequent question type: Differentiate a polynomial, rational, trigonometric, or exponential function; find the equation of the tangent line at a point. ⚡ Exam tip: For implicit differentiation, differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx. Do not forget to apply the chain rule to terms involving y.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Differentiation — CUET Study Guide

Differentiation measures the instantaneous rate of change of a function. The derivative f’(x) = lim_{h→0} [f(x+h) − f(x)]/h measures the slope of the tangent line at x. Geometrically, dy/dx is the slope of the tangent; physically, it is velocity if y is position.

Key formulas to memorise:

  • Polynomial: d/dx(xⁿ) = nxⁿ⁻¹.
  • Trigonometric: d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec² x.
  • Exponential: d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ ln a.
  • Logarithmic: d/dx(ln x) = 1/x; d/dx(logₐ x) = 1/(x ln a).
  • Product rule: (uv)’ = u’v + uv’.
  • Quotient rule: (u/v)’ = (u’v − uv’)/v².
  • Chain rule: (f ∘ g)‘(x) = f’(g(x)) g’(x).

Higher-order derivatives: f”(x) is the derivative of f’(x); f'''(x) is the third derivative. The nth derivative of xⁿ is n! if the order equals n, otherwise zero for terms beyond xⁿ.

Applications:

  • Tangent line: Equation at x = a: y − f(a) = f’(a)(x − a).
  • Increasing/Decreasing: f’(x) > 0 ⇒ function is increasing; f’(x) < 0 ⇒ decreasing.
  • Maxima/Minima: f’(x) = 0 at stationary points. Use second derivative test: f”(x) > 0 ⇒ local minimum; f”(x) < 0 ⇒ local maximum.

Practice Numerical 1: Find dy/dx if y = x² sin x.

  • Using product rule: u = x², v = sin x → u’ = 2x, v’ = cos x.
  • dy/dx = 2x sin x + x² cos x.

Practice Numerical 2: Find the equation of the tangent to y = e^{2x} at x = 0.

  • y(0) = e⁰ = 1. dy/dx = 2e^{2x} → dy/dx|{x=0} = 2.
  • Tangent: y − 1 = 2(x − 0) → y = 2x + 1.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer timeline.

Differentiation — Comprehensive CUET Notes

Deeper theory and proof:

Proof of the chain rule using limits: Let y = f(g(x)). Define the difference quotient Δy/Δx = [f(g(x+Δx)) − f(g(x))]/Δx. As Δx → 0, g(x+Δx) → g(x) (by continuity of differentiable functions). Rewrite the quotient as [f(g(x+Δx)) − f(g(x))]/[g(x+Δx) − g(x)] · [g(x+Δx) − g(x)]/Δx. The first factor tends to f’(g(x)) and the second to g’(x). Hence (f ∘ g)‘(x) = f’(g(x)) g’(x). This formalises the chain rule from first principles.

Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), there exists c ∈ (a, b) such that f’(c) = [f(b) − f(a)]/(b − a). Geometrically, the tangent at c is parallel to the secant joining (a, f(a)) and (b, f(b)). This theorem underlies many results in calculus, including l’Hôpital’s rule and the relationship between increasing/decreasing functions and their derivatives.

l’Hôpital’s Rule: For indeterminate forms 0/0 or ∞/∞, if lim_{x→a} f(x)/g(x) yields an indeterminate form, then lim_{x→a} f(x)/g(x) = lim_{x→a} f’(x)/g’(x), provided the limit on the right exists. Example: lim_{x→0} sin x/x = lim_{x→0} cos x/1 = 1. This is frequently tested in CUET.

Taylor and Maclaurin series: If f is infinitely differentiable at a, f(x) = Σ_{n=0}^{∞} f⁽ⁿ⁾(a)/n! · (x − a)ⁿ. For a = 0, this is the Maclaurin series. Example: eˣ = Σ xⁿ/n!; sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!.

Leibniz rule for higher derivatives of a product: (uv)⁽ⁿ⁾ = Σ_{k=0}^{n} C(n, k) u⁽ᵏ⁾ v⁽ⁿ⁻ᵏ⁾. This generalises the product rule to nth derivatives and is useful for series expansion of product of two functions.

Cross-topic connection — kinematics: Velocity v(t) = ds/dt, acceleration a(t) = dv/dt = d²s/dt². If s(t) = t³ − 6t² + 9t + 1, then v(t) = 3t² − 12t + 9, a(t) = 6t − 12. Setting a(t) = 0 gives t = 2, at which point velocity is v(2) = 3(4) − 12(2) + 9 = 3.

Challenging solved example: Show that the function f(x) = x³ − 3x² + 6x − 4 is strictly increasing on ℝ.

  • Compute f’(x) = 3x² − 6x + 6 = 3(x² − 2x + 2) = 3[(x − 1)² + 1] > 0 for all x ∈ ℝ. Since f’(x) > 0 everywhere, f is strictly increasing on ℝ. The quadratic (x − 1)² + 1 has discriminant Δ = (−2)² − 4(1)(2) = 4 − 8 = −4 < 0, confirming it is always positive.