Chapter 12 — Limits and Derivatives

Limits and Derivatives is an introduction to calculus covering how functions change. Derivatives measure the instantaneous rate of change at a point, computed as the limit of average rates of change.

Chapter 12 introduces calculus, the study of change in function values. Limits establish the foundational concept—the value a function approaches as the input approaches a point. Derivatives quantify instantaneous rates of change, essential for understanding physical phenomena like velocity and acceleration. The chapter covers limit algebra (sum, difference, product, quotient rules), trigonometric limits including the important lim(sin x/x) = 1 as x→0, and derivative computation using first principles. Standard derivatives of polynomials and trigonometric functions follow from the definition f'(x) = lim[f(x+h)−f(x)]/h as h→0, with product and quotient rules enabling complex calculations.

1. 01Limit of f(x) as x→a equals l when both left and right hand limits exist and equal l (denoted lim f(x) = l)
2. 02Derivative f'(a) = lim[f(a+h)−f(a)]/h measures the slope of the tangent to the curve at point a
3. 03Algebra of limits: sums, differences, products, and quotients of limits equal limits of those operations (when denominators ≠ 0)
4. 04Key trigonometric limits: lim(sin x/x) = 1 and lim(1−cos x)/x = 0 as x→0; proven using the sandwich theorem
5. 05Derivative rules: (u+v)' = u'+v', (uv)' = u'v+uv' (product rule), (u/v)' = (u'v−uv')/v² (quotient rule)
6. 06Standard derivatives: (x^n)' = nx^(n−1), (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = sec² x