Chapter 5 at a Glance

Continuity

$f$ is continuous at a domain point $c$ when $\lim_{x \to c} f(x) = f(c)$ — left limit, right limit and value all present and equal. Sums, differences, products and quotients (denominator nonzero) of continuous functions stay continuous, and so do composites; polynomials, rational functions, $\sin$, $\cos$ and $\vert x \vert$ are continuous throughout their domains, while $[x]$ jumps at every integer.

Differentiability

$f'(c) = \lim_{h \to 0}\frac{f(c+h) - f(c)}{h}$, when the limit exists. Every differentiable function is continuous; the converse fails — $\vert x \vert$ has a corner at $0$ (LHD $-1$, RHD $+1$).

The derivative table

The techniques

Chain rule: $\frac{d}{dx}v(u(x)) = v'(u(x))\,u'(x)$ — one factor per layer. Implicit: differentiate the relation; every $y$ triggers a $\frac{dy}{dx}$. Logarithmic: for $u(x)^{v(x)}$ (both positive where needed), $\log y = v\log u$ first. Parametric: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ (with $\frac{dx}{dt} \neq 0$). Second order: differentiate again; for relation proofs, clear radicals/denominators after the first round, then differentiate the cleaned equation.

Mistake checklist

1. Forgetting the inner derivative — $\frac{d}{dx}\sin(x^2) = 2x\cos(x^2)$, never just $\cos(x^2)$.
2. Continuity at joints only — piecewise functions can misbehave only at their joints; test each joint's LHL, RHL and value separately.
3. Differentiable needs continuous first — and $\vert x - a \vert$ is the standing example that continuity alone is not enough.
4. $x^x$ is not $x \cdot x^{x-1}$ — variable-to-variable powers demand logarithmic differentiation; keep the classification straight ($x^a$: power rule; $a^x$: $a^x\log a$).
5. Parametric fractions the right way up — $\frac{dy/dt}{dx/dt}$, and the second derivative is not $\frac{d^2y/dt^2}{d^2x/dt^2}$.
6. Simplify inverse-trig arguments first — the substitution dictionary ($x = \tan\theta$, $x = \sin\theta$, $x = \cos\theta$) turns page-long quotient rules into one-liners.
7. $\frac{d^2y}{dx^2} \neq \left(\frac{dy}{dx}\right)^2$ — the superscript counts repetitions, not powers.

The 15 questions below are a fast pass over the whole chapter at recall level. Each explanation names the section to revisit if it feels shaky.