Revise Differentiation rules: chain, product and quotient for IB Mathematics: Analysis and Approaches (5.2)

Before combining functions, you need the derivatives of the building blocks at your fingertips. The power rule handles every power of $x$, and a short table of standard results covers the trigonometric, exponential and logarithmic functions. These are given facts on the AA SL course — commit them to memory, because the three combination rules all depend on differentiating the pieces correctly first.

Power rule: $\; \dfrac{d}{dx}\big($x^n\big$) = n\,x^{n-1} \quad ($\text{any rational } n$)$

$\dfrac{d}{dx}(\sin x) = \cos x \qquad \dfrac{d}{dx}(\cos x) = -\sin x \qquad \dfrac{d}{dx}(\tan x) = \sec^2 x$

$\dfrac{d}{dx}(e^x) = e^x \qquad \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}$

The chain rule is the most widely used differentiation rule. It applies to composite functions — a function 'nested' inside another — such as $y = \sin(x^2)$, $y = \ln(3x-2)$ or $y = (5x+4)^7$. Differentiate the outer function while keeping the inner function unchanged, then multiply by the derivative of the inner function. That final multiplier is the part students most often forget.

If $y = f(g(x))$, then:

$y' = f'(g(x)) \cdot g'(x)$

Using Leibniz notation, with $y = f(u)$ and $u = g(x)$:

$\dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx}$

The product rule differentiates a function that is the product of two other functions — for example $f(x) = x^3 e^x$ is the product of $u = x^3$ and $v = e^x$. A tempting but wrong shortcut is to multiply the two derivatives; the rule instead accounts for how a change in each factor affects the whole product.

If $f(x) = u(x)\,v(x)$, then:

$f'(x) = u'(x)\,v(x) + u(x)\,v'(x)$

In Leibniz notation, if $y = uv$:

$\dfrac{dy}{dx} = v\,\dfrac{du}{dx} + u\,\dfrac{dv}{dx}$

The quotient rule differentiates a function written as a fraction whose numerator and denominator are both functions of $x$, such as $f(x) = \dfrac{\sin x}{x}$. The order of the two terms in the numerator is fixed and is a frequent source of sign errors.

If $f(x) = \dfrac{u(x)}{v(x)}$, then:

$f'(x) = \dfrac{u'(x)\,v(x) - u(x)\,v'(x)}{[v(x)]^2}$

In Leibniz notation, if $y = \dfrac{u}{v}$:

$\dfrac{dy}{dx} = \dfrac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^2}$

Many exam questions need more than one rule in the same problem — most often a product or quotient where one part itself requires the chain rule. The strategy is always the same: identify the outermost structure first, apply its rule, and reach for a second rule to differentiate whichever piece needs it.

In Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) for a valid approach or an accuracy mark (A) for a correct result. An A-mark is dependent: it is only available once the corresponding method is on the page. Follow-through (FT) means a wrong value carried correctly into later work still earns the later marks, and the engine accepts any equivalent form and ignores correct subsequent working (ISW). Study how each mark below is earned by a specific line in a question that combines the product and chain rules.

These three rules are the engine of the whole calculus strand. Every later technique — finding tangents and normals, locating stationary points and classifying them, optimisation, and even reversing the process in integration by recognising a chain-rule pattern — assumes you can differentiate any AA SL function fluently. Master the habit here: read the structure, pick the rule, differentiate the pieces on their own lines, then combine and simplify. Do that consistently and the calculus that follows becomes variations on a method you already own.