1. Core Concept
Differentiation measures the instantaneous rate of change of a quantity. Geometrically, the derivative f′(x) or dydx represents the exact slope of the tangent line to the curve y = f(x) at any given point.
2. Key Formulas: Standard Derivatives
Algebraic & Exponential
Trigonometric Functions
Memory Trick: Derivatives of all "CO-" functions (cos, cot, csc) are negative!
Inverse Trig Functions
3. Rules of Differentiation
- Parametric: If x = f(t) and y = g(t), then
dydx=dy/dtdx/dt
- Logarithmic: Use for y = [f(x)]g(x). Take ln on both sides → ln(y) = g(x)·ln(f(x)), then differentiate implicitly.
4. Application of Derivatives (AoD)
Rate of Change
dydx represents how y changes with x.
Examples: Velocity v = dsdt, Acceleration a =
dvdt.
Tangents & Normals
- Slope of Tangent (mT): f′(x0)
- Slope of Normal (mN):
-1f′(x0)
- Equation: y - y0 = m(x - x0)
Monotonicity & Extrema
- Increasing: f′(x) > 0
- Decreasing: f′(x) < 0
- Critical Points: Set f′(x) = 0 or find where it is undefined.
If f″(x) < 0 at critical point → Local Maxima
If f″(x) > 0 at critical point → Local Minima
First Derivative Test
- Local Maxima: If f′(x) changes sign from positive to negative as x increases through c.
- Local Minima: If f′(x) changes sign from negative to positive as x increases through c.
- Insight: This is often more reliable than the second derivative test, especially if f″(c) = 0.
5. Conceptual Insights
- Visual Intuition: A derivative is a "magnifying glass" looking at the slope of a curve. At local maxima or minima, the tangent line becomes completely horizontal (slope = f′ = 0).
- Point of Inflection: A point where f″(x) = 0 and the concavity of the curve changes (it doesn't necessarily have to be a maximum or minimum).
When is a function NOT differentiable?
- Sharp corner: The slope changes abruptly (e.g., y = |x| at x = 0).
- Cusp: A point where the tangent becomes vertical and then reverses (e.g., y = x2/3 at x = 0).
- Vertical tangent: The slope becomes infinite (e.g., y = x1/3 at x = 0).
- Discontinuity: A function must be continuous to be differentiable.
6. Common Mistakes
The Parametric Second Derivative Trap
FATAL ERROR:
Correct Method:
- Chain Rule Neglect: Forgetting to differentiate the inner function.
Example: ddxsin(x2) = cos(x2) · 2x (not just cos(x2)).
- Global vs Local: In a closed interval[a, b], the absolute max/min can occur precisely at the endpoints (a, b) even if f′ ≠ 0 there. Always check boundary values!
7. Example Applications
Optimization
To find the largest area of a rectangle inside a circle: Express Area as a single-variable function A(x), find where A′(x) = 0, and verify it's a maximum by checking that A″(x) < 0.
Implicit Differentiation
For a circle equation x2 + y2 = 25, differentiate both sides with
respect to x:
2x + 2y·y′ = 0 → y′ = -x / y.
8. IAT Exam Focus Points
- Rate of Change Problems: Often integrated with 3D geometry or physics (e.g., water leaking from a cone, expanding spherical balloons).
- Increasing/Decreasing Intervals: Finding strict x-ranges where f′(x) > 0 using Wavy Curve / Sign chart methods.
- Tangent/Normal at Parametric Points: Extremely common for Conic Sections (Parabola and Ellipse parameterizations).
- f′(x) = 0 Logic: Using Rolle's Theorem or derivative behavior to definitively find the number of real roots of an algebraic equation.
- L'Hopital's Rule: While technically a limit tool, using derivatives to swiftly solve 0/0 or ∞/∞ forms is essential.
9. Practice Mock Test
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Differentiation & AoD