Calculus Essentials: Derivatives and Their Applications
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Fundamental Concepts of Differentiation
Relationship Between Differentiability and Continuity
If a function f is differentiable at a point x₀, then f is continuous at x₀. However, the converse is not always true; not all continuous functions are differentiable.
Definition of the Derivative at a Point
The derivative of a function f at a point x₀, denoted f'(x₀), is defined as:
f'(x₀) = lim (∆x→0) (f(x₀ + ∆x) - f(x₀))/∆x
Differential of a Function
For a function y = f(x), its differential dy is given by:
dy = y' * dx = f'(x)dx
Invariance of the Differential
The differential df(x) = f'(x)dx
holds true whether x is an independent variable or a function of another variable.
Interpretation of Derivatives and Differentials
The derivative