Module 3: Differentiation Rules
Calculus I – Study Guide
Learn Without Walls
1. Product Rule
d/dx [f(x) · g(x)] = f'(x) g(x) + f(x) g'(x)
Memory: "first-prime times second, plus first times second-prime."
Example: d/dx [x³ ln x] = 3x² ln x + x³(1/x) = 3x² ln x + x².
2. Quotient Rule
d/dx [f(x)/g(x)] = [f'(x) g(x) − f(x) g'(x)] / [g(x)]²
Memory: "low d-high minus high d-low, all over low squared."
Example: d/dx [(x + 1)/(x − 2)] = [(1)(x − 2) − (x + 1)(1)] / (x − 2)² = −3 / (x − 2)².
3. Chain Rule
d/dx [f(g(x))] = f'(g(x)) · g'(x)
Leibniz: dy/dx = (dy/du)(du/dx)
Leibniz: dy/dx = (dy/du)(du/dx)
General Power Rule
d/dx [g(x)]n = n [g(x)]n−1 · g'(x)
Example: d/dx [(2x + 3)5] = 5(2x + 3)4 · 2 = 10(2x + 3)4.
4. Trigonometric Derivatives
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| cot x | −csc² x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
The "co-" functions (cos, cot, csc) all have negative derivatives.
5. Implicit Differentiation
Procedure:
- Differentiate both sides with respect to x.
- Apply the Chain Rule to every y-term: d/dx [f(y)] = f'(y) · dy/dx.
- Collect dy/dx terms on one side, factor, and solve.
Example: x² + y² = 25 ⇒ 2x + 2y(dy/dx) = 0 ⇒ dy/dx = −x/y.
Common Combinations
| Expression | Derivative |
|---|---|
| eg(x) | eg(x) · g'(x) |
| ln(g(x)) | g'(x) / g(x) |
| sin(g(x)) | cos(g(x)) · g'(x) |
| [g(x)]n | n[g(x)]n−1 · g'(x) |