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QLDMath MethodsQuick questions
Unit 4: Further calculus and statistical inference
Quick questions on Implicit differentiation and related rates: QCE Maths Methods Unit 4
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 key idea?Show answer
Treat $y$ as a function of $x$: $y = y(x)$. Differentiate both sides of the equation with respect to $x$, applying the chain rule whenever $y$ appears.
What is worked example. Circle equation?Show answer
Differentiate both sides. $2x + 2y \frac{dy}{dx} = 0$.
What is worked example. Product term?Show answer
Differentiate. $2x + (y + x \frac{dy}{dx}) + 3 y^2 \frac{dy}{dx} = 0$.
What is the four-step procedure?Show answer
Step 1. Identify the variables and relate them. Write an equation linking the time-dependent quantities. Often a geometric formula (volume of a sphere, area of a circle, Pythagoras).
What is standard contexts?Show answer
Sphere inflation. $V = \frac{4}{3} \pi r^3$. $\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$.
What is worked example. Inflating balloon?Show answer
$\frac{dV}{dt} = 50$ cm$^3$/s. Find $\frac{dr}{dt}$ when $r = 10$ cm.
What is worked example. Cone tank?Show answer
A conical tank with height 4 m and top radius 2 m fills with water at 0.5 m$^3$/min. Find the rate at which water depth rises when $h = 1$ m.
What is decreasing rates?Show answer
If a quantity is decreasing, its rate is negative. "Water draining at 5 L/min" gives $\frac{dV}{dt} = -5$. The negative sign carries through the calculation.
What is step 1. Identify the variables and relate them?Show answer
Write an equation linking the time-dependent quantities. Often a geometric formula (volume of a sphere, area of a circle, Pythagoras).
What is step 2. Differentiate both sides with respect to time $t$?Show answer
Use the chain rule on each variable: $\frac{d}{dt}(r^3) = 3 r^2 \frac{dr}{dt}$.
What is sphere inflation?Show answer
$V = \frac{4}{3} \pi r^3$. $\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$.
What is expanding circle?Show answer
$A = \pi r^2$. $\frac{dA}{dt} = 2 \pi r \frac{dr}{dt}$.
What is cone water tank?Show answer
Tank: height $H$, top radius $R$. Water depth $h$, surface radius $r = h R / H$. So $V = \frac{1}{3} \pi r^2 h = \frac{\pi R^2}{3 H^2} h^3$.
What is sliding ladder?Show answer
$x^2 + y^2 = L^2$. $x \frac{dx}{dt} + y \frac{dy}{dt} = 0$. $\frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt}$.
What is shadow length from a walker and a fixed light source?Show answer
Similar-triangle setup; differentiate the linear constraint.