What is a Derivative?
A derivative represents the rate of change of a function at any given point. It tells us how fast a function is changing and in which direction.
Geometric Interpretation: The derivative at a point is the slope of the tangent line to the function's graph at that point.
Physical Interpretation: If f(t) represents position, then f'(t) represents velocity, and f''(t) represents acceleration.
Mathematical Definition: The derivative of f(x) is defined as:
f'(x) = lim[h→0] (f(x+h) - f(x))/h
Basic Differentiation Rules
Power Rule: d/dx(x^n) = n·x^(n-1)
Constant Rule: d/dx(c) = 0
Constant Multiple Rule: d/dx(c·f(x)) = c·f'(x)
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)
Derivatives of Common Functions
Trigonometric Functions:
- d/dx(sin x) = cos x
- d/dx(cos x) = -sin x
- d/dx(tan x) = sec² x
- d/dx(cot x) = -csc² x
- d/dx(sec x) = sec x tan x
- d/dx(csc x) = -csc x cot x
Exponential and Logarithmic Functions:
- d/dx(e^x) = e^x
- d/dx(a^x) = a^x ln(a)
- d/dx(ln x) = 1/x
- d/dx(log_a x) = 1/(x ln a)
Applications of Derivatives
Finding Extrema: Critical points occur where f'(x) = 0 or f'(x) is undefined. Use the second derivative test to determine if these are maxima or minima.
Optimization Problems: Derivatives help find maximum and minimum values in real-world problems.
Related Rates: When two or more quantities change over time, derivatives help relate their rates of change.
Curve Sketching: Derivatives provide information about increasing/decreasing behavior and concavity.
Physics Applications: Position → Velocity → Acceleration relationships in motion problems.