Quick questions on Differentiation rules for HSC Maths Advanced: power, chain, product, quotient, exp, log, trig

The derivative is the gradient of the tangent, and the tangent is the limiting position of a secant. Watching that limit form is the single best way to understand what $f'(x)$ measures. Fix a point $P$ on the curve $y = f(x)$ and let a second point $Q$ slide towards it; the gradient of the line $PQ$ is the average rate of change $\frac{f(a+h) - f(a)}{h}$, and as $h \to 0$ it tends to the instantaneous rate of change at $P$.

The single most common student error is reaching for the wrong rule. Read the structure of the expression first.

Differentiating $f'(x)$ again gives the second derivative $f''(x)$, the rate of change of the gradient. For $f(x) = x^3$, $f'(x) = 3x^2$ and $f''(x) = 6x$. Higher derivatives drive concavity and motion problems, so the rules here feed directly into later calculus topics.

$\frac{d}{dx}(2^x)$ is not $x \cdot 2^{x - 1}$. For non-$e$ exponentials, write $2^x = e^{x \ln 2}$ first, giving $\frac{d}{dx}(2^x) = (\ln 2) \cdot 2^x$.

Markers often reward a clean factored form. After the quotient rule, look for common factors.