Chapter 4: Differentiation
Introduction to Derivatives
The derivative measures how a function changes—its rate of change at a point. Derivatives are central to calculus with applications in optimization, physics, economics, and engineering. This chapter introduces the concept, techniques for computing derivatives, and applications to optimization.
1. The Derivative
Definition
The derivative of f at x, denoted f'(x) or df/dx, is f'(x) = lim(h→0) [f(x+h) - f(x)]/h. This is the instantaneous rate of change of f at x. Geometrically, it's the slope of the tangent line to the curve at that point.
Differentiability
f is differentiable at x if the derivative exists at x. Differentiable functions are continuous, but not all continuous functions are differentiable (e.g., |x| has a sharp corner at x=0, so it's not differentiable there).
2. Differentiation Rules
Power Rule
If f(x) = x^n, then f'(x) = nx^(n-1). For f(x) = x^3, we have f'(x) = 3x^2.
Constant Rule
If f(x) = c (constant), then f'(x) = 0.
Sum and Difference Rule
[f(x) ± g(x)]' = f'(x) ± g'(x). The derivative of a sum is the sum of derivatives.
Product Rule
[f(x)g(x)]' = f'(x)g(x) + f(x)g'(x). Derivative of a product requires both functions and their derivatives.
Quotient Rule
[f(x)/g(x)]' = [f'(x)g(x) - f(x)g'(x)]/[g(x)]^2. More complex than product rule; be careful with signs and order.
Chain Rule
If y = f(g(x)), then dy/dx = (df/dg)(dg/dx). This rule handles composite functions. Example: if y = (x^2 + 1)^5, let u = x^2 + 1, so y = u^5. Then dy/dx = 5u^4 · 2x = 10x(x^2 + 1)^4.
3. Derivatives of Elementary Functions
Exponential and Logarithmic
d/dx(e^x) = e^x. d/dx(ln x) = 1/x. d/dx(a^x) = a^x ln(a) for constant a.
Trigonometric Functions
d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec^2 x.
4. Applications: Maxima and Minima
Critical Points
A critical point is where f'(x) = 0 or f'(x) doesn't exist. These are candidates for local maxima or minima.
First Derivative Test
If f'(x) changes from positive to negative at x = c, then f has a local maximum at c. If f'(x) changes from negative to positive, there's a local minimum.
Second Derivative Test
Compute f''(x). If f'(c) = 0 and f''(c) > 0, then c is a local minimum. If f''(c) < 0, then c is a local maximum. If f''(c) = 0, the test is inconclusive.
Profit Maximization Example
If profit P(x) = 1000x - 5x^2 (revenue minus cost), find maximum profit. P'(x) = 1000 - 10x. Setting P'(x) = 0: x = 100. Check P''(x) = -10 < 0, confirming a maximum. Maximum profit occurs at x = 100 units.
Conclusion
Derivatives quantify change. The rules for differentiation make computing derivatives mechanical once mastered. Applications to optimization are powerful—finding maxima and minima appears in business, engineering, and science constantly. Calculus turns derivative theory into practical problem-solving tools.