Derivative as a Rate Measure — Complete Formula

Basic Idea of Rate of Change

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Average Rate of Change	[\dfrac{f(x_2) – f(x_1)}{x_2 – x_1}]	[f(x)] = function, [x_1, x_2] = points	Over an interval ⭐
Instantaneous Rate	[\dfrac{dy}{dx}]	[y] = dependent variable, [x] = independent variable	At a point ⭐

💡 Memory Hint:
Derivative = instantaneous rate

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Velocity	[v = \dfrac{ds}{dt}]	[s] = displacement, [t] = time	Rate of change of position ⭐
Acceleration	[a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}]	[v] = velocity	Second derivative ⭐
Average Velocity	[\dfrac{s_2 – s_1}{t_2 – t_1}]	—	Over interval
Instantaneous Velocity	[\lim_{\Delta t \to 0} \dfrac{\Delta s}{\Delta t}]	—	Exact velocity ⭐

💡 Memory Hint:
Position → velocity → acceleration

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Increasing	[\dfrac{dy}{dx} > 0]	—	Function rises ⭐
Decreasing	[\dfrac{dy}{dx} < 0]	—	Function falls ⭐
Constant	[\dfrac{dy}{dx} = 0]	—	Flat region

💡 Memory Hint:
Sign of derivative decides behavior

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Chain Rule Form	[\dfrac{dy}{dt} = \dfrac{dy}{dx} \cdot \dfrac{dx}{dt}]	[x, y] = variables depending on [t]	Very important ⭐

💡 Memory Hint:
Differentiate → then substitute values

Concept	Formula	Symbols Meaning	Key Notes
Area Rate	[\dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt}]	[A] = area, [r] = radius	Common JEE ⭐
Circumference Rate	[\dfrac{dC}{dt} = 2\pi \dfrac{dr}{dt}]	[C] = circumference	Direct

Concept	Formula	Symbols Meaning	Key Notes
Volume Rate	[\dfrac{dV}{dt} = 4\pi r^2 \dfrac{dr}{dt}]	[V] = volume	Important ⭐

Concept	Formula	Symbols Meaning	Key Notes
Volume Rate	[\dfrac{dV}{dt} = \pi r^2 \dfrac{dh}{dt} + 2\pi r h \dfrac{dr}{dt}]	[h] = height	Combined change ⭐

💡 Memory Hint:
Differentiate formula → apply chain rule

Concept	Formula	Symbols Meaning	Key Notes
Growth	[\dfrac{dy}{dt} = ky]	[k] = constant	Rate ∝ quantity ⭐
Solution	[y = y_0 e^{kt}]	[y_0] = initial value	Standard model ⭐
Decay	[k < 0]	—	Decreasing quantity

💡 Memory Hint:
Growth → exponential increase

Concept	Formula	Symbols Meaning	Key Notes
Differential	[dy = \dfrac{dy}{dx} dx]	[dx] = small change	Approximation ⭐
Relative Error	[\dfrac{dy}{y}]	—	Accuracy measure

💡 Memory Hint:
Small change → use derivative

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