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QLDMath MethodsQuick questions
Unit 4: Further calculus and statistical inference
Quick questions on Further differentiation and applications: QCE Maths Methods Unit 4
11short 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 procedure?Show answer
1. Take the natural logarithm of both sides. $\ln y = \ln(f(x))$. 2. Simplify using log rules. $\ln(a b) = \ln a + \ln b$; $\ln(a/b) = \ln a - \ln b$; $\ln(a^k) = k \ln a$.
What is worked example?Show answer
Differentiate. $\frac{1}{y} \frac{dy}{dx} = \ln x + 1$ (product rule on $x \ln x$).
What is when to use it?Show answer
Logarithmic differentiation is the right tool when:
What is worked example. Inverse of $f(x) = x^3 + x$?Show answer
$f(x) = x^3 + x$. Find the derivative of $f^{-1}$ at $x = 2$.
What is optimisation with logarithmic differentiation?Show answer
The revenue from selling $x$ items is $R(x) = x \cdot p(x)$ where $p(x) = 100 - x^{0.5}$. Find $x$ that maximises $R$.
What is implicit derivative for shape constraints?Show answer
If a curve is defined implicitly by $x^2 + y^2 + xy = 10$, find $dy/dx$ at the point $(1, 2)$ (which satisfies the equation: $1 + 4 + 2 = 7$ - let me re-check: $(1)^2 + (2)^2 + (1)(2) = 1 + 4 + 2 = 7 \neq 10$. Use $(2, 1)$: $4 + 1 + 2 = 7$, still not. Skip the worked specific point; use general approach).
What is forgetting the chain rule inside?Show answer
Differentiating $\ln(x^2 + 1)$ requires the chain rule: $\frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}$.
What is logarithmic differentiation without final $y$ substitution?Show answer
After computing $\frac{1}{y} \frac{dy}{dx}$, you must multiply by $y$ and replace $y$ with the original expression.
What is confusing $\frac{d}{dx} $ and $\frac{d}{dx} $?Show answer
The natural log has derivative $1/x$. The base-10 log has derivative $1 / (x \ln 10)$. QCAA Methods uses natural log throughout.
What is power rule applied to variable exponent?Show answer
$\frac{d}{dx}(x^x)$ is NOT $x \cdot x^{x-1} = x^x$. The power rule applies only to constant exponents. Use logarithmic differentiation.
What is sign error in inverse function rule?Show answer
The derivative of $f^{-1}$ is $1 / f'(y)$, not $1 / f'(x)$. The $y$ is the value of the inverse function at $x$.