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Introduction
Rate of change tells how one quantity changes with respect to another quantity. In calculus, the derivative \(\frac{dy}{dx}\) represents the rate of change of \(y\) with respect to \(x\), and this idea is used in geometry, motion, business mathematics, and many real-life situations.
Definition: Rate of Change
If a quantity y varies with another quantity x based on a rule y = f(x), then the derivative \[\frac{dy}{dx}\] (or f'(x)) represents the rate of change of y with respect to x.
Definition: Specific Instant
Evaluating the derivative at a specific point, \[\left.\frac{dy}{dx}\right|_{x=x_0}\], gives the instantaneous rate of change at exactly \[x = x_0\].
Definition: The Chain Rule for Rates
If two variables x and y both vary with respect to a third variable t (like time), you can find the rate of change of y with respect to x using:
(Note: This is only valid if \[\frac{dx}{dt} \neq 0\]).
Example 1
Find the rate of change of the area of a circle with respect to its radius \[r\] when \[r = 5\] cm.
Solution: The area A of a circle with radius \[r\] is given by \[\text{A} = \pi r^2\]. Therefore, the rate of change of the area A with respect to its radius \[r\] is given by \[\frac{d\text{A}}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r\].
When \[r = 5\] cm, \[\frac{d\text{A}}{dr} = 10\pi\]. Thus, the area of the circle is changing at the rate of \[10\pi\] cm\[^2\]/s.
Example 2
The total cost \[\text{C}(x)\] in Rupees, associated with the production of \[x\] units of an item is given by
Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.
Solution: Since marginal cost is the rate of change of total cost with respect to the output, we have
Marginal cost (MC) \[= \frac{d\text{C}}{dx} = 0.005(3x^2) - 0.02(2x) + 30\]
When \[x = 3, \text{MC} = 0.015(3^2) - 0.04(3) + 30\]
\[= 0.135 - 0.12 + 30 = 30.015\]
Hence, the required marginal cost is ₹ 30.02 (nearly).
Key Points: Rate of Change of Quantities
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Derivative gives instantaneous rate of change.
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Positive derivative means the quantity is increasing.
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Negative derivative means the quantity is decreasing.
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In related rates, first connect the variables by an equation, then differentiate.
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Always substitute the given value only after differentiation.
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Do not forget units in the final answer.
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Marginal cost and marginal revenue are applications of derivatives in economics.
