Calculus I Foundations (MATH 151) · Module 3 of 11
The Derivative: Definition & Basic Rules
The formal definition of the derivative, and the shortcut rules that make differentiating fast.
The Definition of the Derivative
- f'(x) = lim(h→0) [f(x+h) − f(x)] / h, the instantaneous rate of change of f at x, or the slope of the tangent line at that point.
What does the derivative f'(a) represent geometrically?
Solution. The slope of the tangent line to the graph of f at the point (a, f(a)).
Power, Constant, and Sum Rules
- Power rule: d/dx[xⁿ] = n·xⁿ⁻¹. Constant multiple rule: d/dx[c·f(x)] = c·f'(x). Sum/difference rule: d/dx[f(x) ± g(x)] = f'(x) ± g'(x).
Find the derivative of f(x) = 3x⁴ − 5x + 7.
Solution. f'(x) = 3(4x³) − 5(1) + 0 = 12x³ − 5.
Product & Quotient Rules
- 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.
Find the derivative of f(x) = x²·sin(x), using the product rule.
Solution. f'(x) = (2x)(sin x) + (x²)(cos x) = 2x sin x + x² cos x.
Derivatives of Trigonometric Functions
- d/dx[sin x] = cos x; d/dx[cos x] = −sin x; d/dx[tan x] = sec²x.
Find the derivative of f(x) = cos(x).
Solution. f'(x) = −sin(x).
Check yourself
2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.
1. What is the definition of the derivative f'(x)?
2. What is the derivative of x⁵ using the power rule?