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QLDMath MethodsQuick questions
Unit 3: Further calculus and statistics
Quick questions on Product, quotient and chain rules in combination (QCE Mathematical Methods Unit 3)
15short 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 chain rule?Show answer
If $y = f(g(x))$, set $u = g(x)$ so $y = f(u)$. Then
What is the product rule?Show answer
$$\frac{dy}{dx} = u' v + u v'.$$
What is the quotient rule?Show answer
If $y = \dfrac{u(x)}{v(x)}$ with $v(x) \neq 0$, then
What is standard derivatives (the library)?Show answer
The rules above act on the standard derivatives. Memorise these.
What is order of operations?Show answer
When two or more rules combine, choose the outer structure first and work inwards.
What is chain inside a product?Show answer
Differentiate $y = x \sin(2x)$.
What is chain inside a quotient?Show answer
Differentiate $y = \dfrac{e^{2x}}{x + 1}$.
What is double chain?Show answer
Differentiate $y = \sin^2(3x)$, that is $y = (\sin(3x))^2$.
What is logarithm with the product rule?Show answer
Product rule with $u = x$, $v = \ln x$, $u' = 1$, $v' = \tfrac{1}{x}$.
What is quotient with a chain inside?Show answer
Differentiate $y = \dfrac{\cos(3x)}{x^2}$.
What is forgetting the chain rule on composed functions?Show answer
Writing $\frac{d}{dx} \sin(2x) = \cos(2x)$ drops the factor of $2$. The correct answer is $2 \cos(2x)$.
What is reversing the quotient rule sign?Show answer
The numerator is $u' v$ minus $u v'$. Writing it the other way around flips the sign of the entire derivative.
What is treating $e^{2x}$ like a power?Show answer
$\frac{d}{dx}(e^{2x}) = 2 e^{2x}$, by the chain rule. It is not $2x \, e^{2x - 1}$.
What is applying the power rule to $a^x$?Show answer
For non-$e$ exponentials, $\frac{d}{dx}(a^x) = (\ln a) \, a^x$. The power rule does not apply because the base is constant and the exponent is variable.
What is not simplifying?Show answer
QCAA frequently allocates a mark for a clean final form. After a product or quotient rule, look for common factors and factor them out.