1. Limit, Continuity & Differentiability
Understanding limits helps us analyze how a function behaves as it approaches a specific point. Continuity ensures that a function doesn't have any breaks, and differentiability indicates that the function has a well-defined tangent at that point.
- Limit: The value a function approaches as the input approaches a certain point.
- Continuity: A function is continuous at a point if the limit exists, the function is defined at that point, and the value equals the limit.
- Differentiability: A function is differentiable at a point if it has a finite derivative at that point (i.e., smooth without sharp corners).
2. Rules of Differentiation
Differentiation follows certain rules that make the process systematic:
- Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
- Product Rule: d/dx [f(x)·g(x)] = f'(x)g(x) + f(x)g'(x)
- Quotient Rule: d/dx [f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x))/g(x)^2
- Chain Rule: d/dx f(g(x)) = f'(g(x)) · g'(x)
3. Applications of Derivatives
Derivatives have numerous real-world applications. Some key uses include:
- Rate of Change: Represents how one quantity changes with respect to another (e.g., velocity is the rate of change of position).
- Tangents: The derivative at a point gives the slope of the tangent to the curve at that point.
- Normals: The normal is a line perpendicular to the tangent at a given point. Its slope is the negative reciprocal of the tangent's slope.
Example: For f(x) = x², the derivative is f'(x) = 2x. At x = 3, the slope of the tangent is 6 and the slope of the normal is -1/6.