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Differentiation and Its Applications

Quick questions on Connected rates of change explained: O-Level A-Maths

7short 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 link?
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If yy depends on xx, and both change with time tt, the chain rule connects their rates:
What is rearranging for the unknown rate?
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The chain rule can be rearranged to isolate whichever rate you need. To find drdt\dfrac{dr}{dt} from dVdt\dfrac{dV}{dt}:
What is reading the question for the given rate?
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The phrase "increases at" or "decreases at" gives you a time-rate directly. A decrease is a negative rate. Match the units to confirm which rate is given and which is wanted.
What are mismatched units?
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Keep units consistent so the final rate carries the correct units, such as cm2 s−1\text{cm}^2\,\text{s}^{-1} for an area rate.
What is q1?
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Given A=πr2A = \pi r^2, write dAdr\dfrac{dA}{dr}. [1 mark]
What is q2?
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The radius of a circle grows at 0.5 cm s−10.5\ \text{cm s}^{-1}. Find dAdt\dfrac{dA}{dt} when r=4 cmr = 4\ \text{cm}. [3 marks]
What is q3?
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A cube's side increases at 2 cm s−12\ \text{cm s}^{-1}. Find the rate of increase of its volume when the side is 5 cm5\ \text{cm}. [3 marks]

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