Revise Further differentiation and the second derivative for IB Mathematics: Applications and Interpretation (5.5)

Before the combination rules you need the derivatives of the individual functions. These four are on your formula booklet, but you should know them cold, along with the crucial detail that the derivative of $\cos x$ carries a minus sign and that the trig results assume radians.

$\dfrac{d}{dx}e^{x}=e^{x}$

$\dfrac{d}{dx}\ln x=\dfrac{1}{x}$

$\dfrac{d}{dx}\sin x=\cos x$

$\dfrac{d}{dx}\cos x=-\sin x$

Real functions are usually built from these standard pieces. The chain rule handles a function nested inside another, like $e^{x^2}$ or $\sin(3x)$: differentiate the outer function, keep the inner one unchanged, then multiply by the derivative of the inner function. The product rule handles two functions multiplied, like $x^2\sin x$. The quotient rule handles a fraction, like $\frac{\ln x}{x}$ — and there the order in the numerator matters because of the minus sign.

Chain rule: if $y=f(g(x))$ then $\dfrac{dy}{dx}=f'(g(x))\cdot g'(x)$.

Product rule: if $y=uv$ then $\dfrac{dy}{dx}=u'v+uv'$.

Quotient rule: if $y=\dfrac{u}{v}$ then $\dfrac{dy}{dx}=\dfrac{u'v-uv'}{v^{2}}$.

Differentiate $f(x)$ and you get $f'(x)$, the gradient. Differentiate again and you get the second derivative $f''(x)$ — also written $\frac{d^2y}{dx^2}$ — which measures how the gradient itself is changing. Where the gradient is increasing the curve bends upward; where it is decreasing the curve bends downward. That bending is called concavity, and its sign is read directly off $f''$.

The second derivative gives the quickest way to classify a stationary point. At a stationary point the tangent is horizontal, so the curve is turning. If it is concave down there ($f''<0$) it must be turning at a peak — a local maximum. If it is concave up there ($f''>0$) it is turning at a trough — a local minimum. You only evaluate $f''$ at a single point, which is why this test is usually faster than a first-derivative sign check.

In an exam the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if the arithmetic later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct step performed on your own earlier (wrong) value still scores, and equivalent correct forms are accepted. All of that protection exists only if your method is on the page. Study how each mark below is earned by a specific line.

Your GDC checks all of this analytic work. Graph $f$ and read off maxima and minima directly, or use the maximum/minimum finder over a sensible interval. The numerical derivative feature gives $f'(a)$ at a point without algebra, and applying it to $f'$ (or graphing $f'$ and finding where it turns) locates where $f''=0$, i.e. candidate inflexion points. Keep the calculator in radian mode for any trig differentiation, and use the graph shape to confirm concavity — cup-shaped where $f''>0$, cap-shaped where $f''<0$.

You now have the full HL differentiation toolkit: the standard functions, the three combination rules, and the second derivative for reading curvature and classifying turning points. This is the machinery behind the modelling questions on Paper 2 — optimising a real quantity, describing where a model accelerates or decelerates, finding where a rate of change is greatest — and it underpins the reverse process, integration, which comes next. Master the habit — differentiate, set $f'=0$, then use $f''$ to classify and to read the shape — and the rest of HL calculus becomes variations on a method you already own.