Additional Mathematics 4037/22 — May/June 2019

Walkthrough

The function is a quotient of two differentiable functions, so the quotient rule is the natural tool. We take $u = \sin x$ and $v = \ln x^2$.

First find each derivative separately. The derivative of $\sin x$ is simply $\cos x$. For the denominator, $\ln x^2$ is a composite function: the outer function is $\ln(\cdot)$ and the inner is $x^2$, so the chain rule gives $\frac{1}{x^2} \times 2x = \frac{2}{x}$. (Equivalently, $\ln x^2 = 2\ln x$ for $x > 0$, whose derivative is also $\frac{2}{x}$.)

Then substitute both derivatives into the quotient rule $\frac{v u' - u v'}{v^2}$, giving numerator $(\ln x^2)\cos x - (\sin x)\frac{2}{x}$ over $(\ln x^2)^2$. No further simplification is required — the mark scheme allows any equivalent form and ignores later mis-simplification.

Key Takeaways

• The quotient rule: if $y = \frac{u}{v}$ then $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{v\frac{\mathrm{d}u}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}$ — order matters in the numerator.
• Differentiating $\ln$ of a composite function requires the chain rule; recognising $\ln x^2 = 2\ln x$ is a useful shortcut.
• Standard derivatives ($\sin x \to \cos x$) should be immediate recall.

Common Mistakes

• Reversing the order of the numerator of the quotient rule (writing $uv' - vu'$), which loses the accuracy mark.
• Writing $\frac{\mathrm{d}}{\mathrm{d}x}(\ln x^2) = \frac{1}{x^2}$ — forgetting the chain-rule factor $2x$.
• Forgetting to square the denominator, i.e. writing only $\ln x^2$ underneath.
• Sign slips when subtracting the second term of the numerator.
• Using the product rule incorrectly if rewriting as $(\sin x)(\ln x^2)^{-1}$ — the derivative of $(\ln x^2)^{-1}$ must include the chain factor $-(\ln x^2)^{-2} \times \frac{2}{x}$.

Things to Be Careful About

• Each individual derivative earns a mark even before the rule is applied, so state $\cos x$ and $\frac{2}{x}$ explicitly.
• Any equivalent form of the final answer is accepted ('oe'), and later mis-simplification is ignored ('isw') — but the correct structure must come from correct working ('nfww').
• Keep the answer exact; no decimals are involved here.