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NSWMaths AdvancedQuick questions
Year 12: Calculus
Quick questions on Differentiation rules for HSC Maths Advanced: power, chain, product, quotient, exp, log, trig
14short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is the power rule?Show answer
For any real $n$, $$\frac{d}{dx}(x^n) = n x^{n - 1}.$$
What is the chain rule?Show answer
If $y = f(g(x))$, let $u = g(x)$ so $y = f(u)$. Then $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$$
What is the product rule?Show answer
If $y = u(x) v(x)$, then $$\frac{dy}{dx} = u' v + u v'.$$
What is the quotient rule?Show answer
If $y = \frac{u(x)}{v(x)}$, then $$\frac{dy}{dx} = \frac{u' v - u v'}{v^2}.$$
What is standard derivatives?Show answer
Memorise these. They appear in nearly every paper.
What is chain rule with trig?Show answer
Differentiate $y = \sin(3x^2)$.
What is product rule?Show answer
$u = x$, $v = e^x$, so $u' = 1$ and $v' = e^x$.
What is quotient rule?Show answer
Differentiate $y = \frac{\ln x}{x}$.
What is combined chain and product?Show answer
Differentiate $y = x^2 \sin(2x)$.
What is forgetting the chain rule on composed functions?Show answer
Writing $\frac{d}{dx}(\sin(2x)) = \cos(2x)$ loses the factor of $2$. The correct answer is $2 \cos(2x)$.
What is mixing up the quotient rule sign?Show answer
The numerator is $u' v$ minus $u v'$, in that order. Reversing it changes the sign.
What is treating $e^{2x}$ like $2 e^{2x \log e}$?Show answer
Just use the chain rule: $\frac{d}{dx}(e^{2x}) = 2 e^{2x}$.
What is power rule on $a^x$?Show answer
$\frac{d}{dx}(2^x)$ is not $x \cdot 2^{x - 1}$. For non-$e$ exponentials, write $2^x = e^{x \ln 2}$ first, giving $\frac{d}{dx}(2^x) = (\ln 2) \cdot 2^x$.
What is not simplifying?Show answer
Markers often reward a clean factored form. After the quotient rule, look for common factors.