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Ch 06 - Application of Derivatives12th Mathematics

Derivative as a Rate Measurer

CONCEPT Formula

\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}

Mathematical representation for the instantaneous rate of change.

Derivation

Derived from the First Principle of derivatives, tracking the limit of the average rate of change as the interval shrinks to zero.

Component Anatomy

$\frac{dy}{dx}$ or $f'(x)$ : The instantaneous rate of change of quantity $y$ strictly with respect to quantity $x$ .
$\left. \frac{dy}{dx} \right|_{x=x_0}$ : The specific rate of change evaluated precisely at the instant $x = x_0$ .

Boundary Conditions / Domain

The limit defining the derivative must exist and evaluate to a finite real number.
A positive derivative indicates $y$ strictly increases as $x$ increases. A negative derivative indicates $y$ strictly decreases as $x$ increases.

Prism Memory Hook

Derivative = Rate of Change.

The Chain-Rule Bridge for Related Rates

APPLICATION Formula

\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}

Chain Rule formula connecting the time-rates of two related dynamic variables.

Component Anatomy

$\frac{dy}{dt}$ : Rate of change of dependent variable $y$ w.r.t time $t$ .
$\frac{dx}{dt}$ : Rate of change of independent variable $x$ w.r.t time $t$ .
$\frac{dy}{dx}$ : The geometric or physical mathematical link between $y$ and $x$ (e.g., Area and radius).

Boundary Conditions / Domain

The function connecting $y$ and $x$ must be strictly differentiable.
$\frac{dx}{dt} \neq 0$ if rearranging to isolate and solve for $\frac{dy}{dx}$ .

Prism Memory Hook

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Related Rates: Execution SOP

What is the strict step-by-step Standard Operating Procedure (SOP) for solving Related Rates problems?

PROCEDURAL

Algorithmic SOP

Model: Identify the geometric or physical equation relating the variables (e.g., $A = \pi r^2$ ).
Differentiate: Differentiate the entire relation implicitly strictly with respect to time $t$ .
Substitute: CRITICAL: Substitute the instantaneous numerical values for variables only after differentiating, never before (as plugging in constants prematurely forces their derivative to zero).

Boundary Conditions / Domain

Variables substituted in the final step must correspond to the exact same instant in time $t$ .

Prism Memory Hook

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Also in this chapter

48 more cards to study in Prism

4Shadow Tip Velocity: Formulation
5Shadow Tip Velocity: Geometric Derivation SOP
6Marginal Cost and Marginal Revenue
7Strictly Increasing vs. Increasing Functions
8Monotonicity Theorem via First Derivative
9Monotonic Intervals: Critical Point Phase
10Monotonic Intervals: Sign Evaluation Phase
11Open vs. Closed Intervals in Monotonicity
12The Two Classifications of Critical Points
13The False Converse of Zero Derivative
14Extrema at Non-Differentiable Corners
15Slopes of Tangent and Normal Lines
16Horizontal and Vertical Tangent Conditions
17Implicit Tangents: Differentiation Phase
18Implicit Tangents: Evaluation Phase
19Orthogonal Intersection Condition
20Orthogonal Intersection: Execution SOP
21Normal to Parametric Involute Curve
22Local vs. Absolute (Global) Extrema
23First Derivative Test: Local Maximum
24First Derivative Test: Local Minimum
25Point of Inflection Analysis
26Second Derivative Test for Local Extrema
27Test Selection: 1st vs 2nd Derivative
28Extreme Value Theorem (EVT)
29Absolute Extrema Trap on Open Intervals
30Absolute Extrema: Critical Point Extraction
31Absolute Extrema: Boundary Finalization
32The Second Derivative Trap for Absolute Extrema
33Applied Optimisation: Modeling Phase
34Applied Optimisation: Calculus Phase
35Trigonometric Substitution Heuristic
36Optimising Profit: Marginal Setup Phase
37Optimising Profit: Critical Evaluation Phase
38The 'Wire Cut' Minimum Area Heuristic
39Maximum Area of an Inscribed Rectangle
40Maximum Cylinder Inscribed in a Sphere
41Maximum Cone Inscribed in a Sphere
42Maximum Cylinder Inscribed in a Cone
43Cone Optima (Fixed Slant Height): Parametric Setup
44Cone Optima: Optimal Angle for Fixed Slant Height
45Cone Optima: Given Fixed Surface Area
46Maximum of the Logarithmic Ratio
47Extrema of Sine + Cosine: Evaluation Phase
48Maximum of Sine + Cosine Identity
49Linear Approximation (Differentials)
50Exact Change vs Differential Approximation
51Relative Error Propagation Formula

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