Rate of Change of Quantities

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.

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.

Evaluating the derivative at a specific point, \[\left.\frac{dy}{dx}\right|_{x=x_0}\], gives the instantaneous rate of change at exactly \[x = x_0\].

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 \[\frac{dx}{dt} \neq 0\]).

• Step 1: Identify the quantities changing.
• Step 2: Write the relation between them.
• Step 3: Differentiate with respect to the required variable or time.
• Step 4: Substitute the given values.
• Step 5: Write the answer with correct units.

Find the rate of change of the area of a circle with respect to its radius \[r\] when \[r = 5\] cm.

Solution: 

Area of circle: \[ A = \pi r^2 \]

Differentiating with respect to r, \[ \frac{dA}{dr} = 2\pi r \]

At r = 5,

\[ \frac{dA}{dr} = 2\pi(5) = 10\pi \]

Therefore, the rate of change of area with respect to radius is

\[ \boxed{10\pi \text{ cm}^2/\text{cm}} \]

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 \[ \boxed{MC \approx \text{₹}30.02} \].

• Derivative gives instantaneous rate of change.

• Positive derivative means the quantity is increasing.

• Negative derivative means the quantity is decreasing.

• In related rates, first connect the variables by an equation, then differentiate.

• Always substitute the given value only after differentiation.

• Do not forget units in the final answer.

• Marginal cost and marginal revenue are applications of derivatives in economics.