This topic covers the differentiation and integration of the full range of functions encountered in Specialist Mathematics, together with the standard techniques applied to combine and transform them.
| Rule | Formula |
|---|---|
| Product | $(uv)’ = u’v + uv’$ |
| Quotient | $(u/v)’ = (u’v - uv’)/v^2$ |
| Chain | $(f \circ g)’(x) = f’(g(x)) \cdot g’(x)$ |
| Implicit | Differentiate both sides w.r.t. $x$; treat $y$ as a function of $x$ |
| $f(x)$ | $f’(x)$ |
|---|---|
| $x^n$ | $nx^{n-1}$ |
| $e^{ax}$ | $ae^{ax}$ |
| $\ln(ax)$ | $1/x$ |
| $\sin(ax)$ | $a\cos(ax)$ |
| $\cos(ax)$ | $-a\sin(ax)$ |
| $\tan(ax)$ | $a\sec^2(ax)$ |
| $\arcsin(x)$ | $1/\sqrt{1-x^2}$ |
| $\arccos(x)$ | $-1/\sqrt{1-x^2}$ |
| $\arctan(x)$ | $1/(1+x^2)$ |
| $f(x)$ | $\int f(x)\,dx$ |
|---|---|
| $x^n$ $(n\neq-1)$ | $x^{n+1}/(n+1) + c$ |
| $1/x$ | $\ln |
| $e^{ax}$ | $e^{ax}/a + c$ |
| $\sin(ax)$ | $-\cos(ax)/a + c$ |
| $\cos(ax)$ | $\sin(ax)/a + c$ |
| $\sec^2(ax)$ | $\tan(ax)/a + c$ |
| $1/\sqrt{a^2-x^2}$ | $\arcsin(x/a) + c$ |
| $1/(a^2+x^2)$ | $(1/a)\arctan(x/a) + c$ |
Substitution (reverse chain rule): Replace a composite expression with $u$.
Example: $\displaystyle\int 2x e^{x^2}\,dx$. Let $u = x^2$, $du = 2x\,dx$:
$$\int e^u\,du = e^u + c = e^{x^2} + c$$
Integration by parts: $\displaystyle\int u\,dv = uv - \int v\,du$.
Choose $u$ to differentiate (algebraic, then logarithmic, then other — LIATE rule).
Example: $\displaystyle\int x e^x\,dx$. Let $u = x$, $dv = e^x\,dx$, so $du = dx$, $v = e^x$:
$$xe^x - \int e^x\,dx = xe^x - e^x + c = e^x(x-1) + c$$
Partial fractions: For rational integrands (see Partial Fractions notes).
| Integrand form | Technique |
|---|---|
| $f’(x)/f(x)$ | $\ln |
| $f’(x)\cdot[f(x)]^n$ | Substitution |
| Product: polynomial $\times$ exp/trig | Integration by parts |
| Product: $\ln x$ or $\arctan x$ times something | Parts with $u = \ln x$ or $\arctan x$ |
| Rational function | Partial fractions |
| $1/(a^2+x^2)$ or $1/\sqrt{a^2-x^2}$ | Inverse trig formula |
KEY TAKEAWAY: Advanced calculus is the engine of Specialist Mathematics. A fluent, confident command of the differentiation rules and integration techniques is required for both paper 1 (no CAS) and paper 2.
STUDY HINT: Build a personalised table of standard results and review it daily. Paper 1 provides a formula sheet, but being able to recall rules instantly saves precious time.
VCAA FOCUS: Integration by parts with logarithmic or inverse-trig functions, and substitution into trig or exponential forms, regularly appear in Paper 2 extended-answer sections.
