Standard derivatives
Continuity and Differentiability · Differentiability and the Chain Rule\[(x^n)'=nx^{n-1},\ (\sin x)'=\cos x,\ (\cos x)'=-\sin x,\ (\tan x)'=\sec^2x\]\[(\sec x)'=\sec x\tan x,\ (\csc x)'=-\csc x\cot x,\ (\cot x)'=-\csc^2x\]
Chapter 5 · Calculus
Continuity and Differentiability
Chapter Overview
Calculus studies change. Before we can measure how fast something changes, we need functions without sudden jumps (continuity) and without sharp corners (differentiability). This chapter makes both ideas precise and then builds a toolkit for differentiating almost any function you meet: the chain rule, implicit differentiation, derivatives of inverse trigonometric, exponential and logarithmic functions, logarithmic differentiation, parametric forms and second derivatives.
Why it matters: Calculus carries 35 of the 80 board marks. Every technique here is used again in Applications of Derivatives, Integrals and Differential Equations.
Topics
Key Concepts
- \(f\) is continuous at \(c\) if \(\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c)\).
- Differentiable at \(c\) ⇒ continuous at \(c\); the converse is false (e.g. \(|x|\) at 0).
- Chain rule: \(\frac{d}{dx}f(g(x))=f\,'(g(x))\,g'(x)\).
- Logarithmic differentiation handles \(u^v\) and long products; parametric: \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\).
Formulas
Chain rule
Continuity and Differentiability · Differentiability and the Chain Rule\[\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\]
Inverse trigonometric derivatives
Continuity and Differentiability · Derivatives of Inverse Trigonometric Functions\[(\sin^{-1}x)'=\frac{1}{\sqrt{1-x^2}},\ (\tan^{-1}x)'=\frac1{1+x^2},\ (\sec^{-1}x)'=\frac{1}{|x|\sqrt{x^2-1}}\]
Exponential and logarithm
Continuity and Differentiability · Exponential and Logarithmic Functions\[(e^x)'=e^x,\ (a^x)'=a^x\log a,\ (\log x)'=\frac1x\]
Variable power
Continuity and Differentiability · Logarithmic Differentiation\[\frac{d}{dx}\,u^v=u^v\left(v'\log u+\frac{v\,u'}{u}\right)\]
Parametric derivative
Continuity and Differentiability · Derivatives of Parametric Functions\[\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\]
Key Points
Continuity test
Continuity and Differentiability · ContinuityLHL = RHL = \(f(c)\). For piecewise functions, test only the joining points.
Continuous but not differentiable
Continuity and Differentiability · Differentiability and the Chain Rule\(|x-c|\) at \(c\); \(x^{1/3}\) at 0 (vertical tangent). Differentiability ⇒ continuity, never the reverse.
Substitution shortcuts
Continuity and Differentiability · Derivatives of Inverse Trigonometric FunctionsRecognise multiple-angle formulas: with \(x=\tan\theta\), \(\frac{2x}{1-x^2}=\tan2\theta\) and \(\frac{1-x^2}{1+x^2}=\cos2\theta\); with \(x=\sin\theta\), \(3x-4x^3=\sin3\theta\). Always check that the resulting angle lies in the principal branch.
Common Mistakes
Forgetting the inner derivative
Continuity and Differentiability · Differentiability and the Chain Rule\(\frac{d}{dx}\sin(3x)=3\cos(3x)\), not \(\cos(3x)\).
Missing dy/dx on y-terms
Continuity and Differentiability · Derivatives of Implicit Functions\(\frac{d}{dx}(y^3)=3y^2\frac{dy}{dx}\), and \(\frac{d}{dx}(xy)=y+x\frac{dy}{dx}\).
Treating xˣ as a power or an exponential
Continuity and Differentiability · Logarithmic Differentiation\(\frac{d}{dx}x^x\neq x\cdot x^{x-1}\) and \(\neq x^x\log x\). Correct: \(x^x(1+\log x)\).
Second derivative in parametric form
Continuity and Differentiability · Derivatives of Parametric Functions\(\frac{d^2y}{dx^2}\neq\frac{d^2y/dt^2}{d^2x/dt^2}\). Differentiate \(\frac{dy}{dx}\) with respect to \(t\), then divide by \(\frac{dx}{dt}\).
Solved Examples
Making a function continuous
Continuity and Differentiability · ContinuityFind \(k\) so that \(f(x)=\begin{cases}\frac{\sin 5x}{x},&x\neq0\\k,&x=0\end{cases}\) is continuous at 0.
\(\lim_{x\to0}\frac{\sin5x}{x}=5\lim\frac{\sin5x}{5x}=5\), so \(k=5\).
Differentiating (sin x)ˣ
Continuity and Differentiability · Logarithmic Differentiation\(y=(\sin x)^x\).
\(\log y=x\log\sin x\Rightarrow\frac{y'}{y}=\log\sin x+x\cot x\Rightarrow y'=(\sin x)^x(\log\sin x+x\cot x)\).
Verifying y″ + y = 0
Continuity and Differentiability · Second Order Derivatives\(y=3\cos x-2\sin x\).
\(y'=-3\sin x-2\cos x,\ y''=-3\cos x+2\sin x=-y\). Hence \(y''+y=0\).
Practice & Tests
Chapter Practice
Untimed mixed questions from all topics. Answers are revealed after you submit.
Start PracticeContinuity and Differentiability – Chapter Test
A board-style chapter test drawn fresh from the question bank each time: MCQs, assertion-reason, numericals, written answers you self-check against model solutions, and a case study.
Start Test