The Comprehensive Guide to Differential & Integral Calculus
Calculus is the mathematics of change and motion. Developed independently by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, calculus is structured around two inverse operations: differentiation (measuring rates of change and tangent slopes) and integration (measuring accumulation and the area enclosed beneath curves).
The Core Differentiation Rules
| Differentiation Rule | Formal Mathematical Definition | Worked Example |
|---|---|---|
| Power Rule | d/dx [xⁿ] = n · xⁿ⁻¹ | d/dx [x⁴] = 4x³; d/dx [3x²] = 6x |
| Constant Multiple Rule | d/dx [c · f(x)] = c · f'(x) | d/dx [7 · sin(x)] = 7 · cos(x) |
| Sum & Difference Rule | d/dx [f(x) ± g(x)] = f'(x) ± g'(x) | d/dx [x³ + 2x] = 3x² + 2 |
| Product Rule | d/dx [f(x) · g(x)] = f'(x)g(x) + f(x)g'(x) | d/dx [x² · eˣ] = 2x eˣ + x² eˣ |
| Quotient Rule | d/dx [f/g] = (f'g − fg') / g² | d/dx [x / (x+1)] = 1 / (x+1)² |
| Chain Rule | d/dx [f(g(x))] = f'(g(x)) · g'(x) | d/dx [(2x + 1)³] = 3(2x + 1)² · 2 = 6(2x + 1)² |
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) demonstrates that differentiation and definite integration are inverse mathematical operations:
Where $F(x)$ is any antiderivative of $f(x)$ such that $F'(x) = f(x)$. For polynomials: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
How Critical Points and Concavity Are Determined
- Critical Points: Points in the domain where $f'(x) = 0$ or $f'(x)$ is undefined. These represent horizontal tangent lines and potential local extrema.
- Second Derivative Test:
- If $f'(c) = 0$ and $f''(c) > 0$: The curve is concave upward ($\cup$), indicating a Local Minimum.
- If $f'(c) = 0$ and $f''(c) < 0$: The curve is concave downward ($\cap$), indicating a Local Maximum.
- If $f''(c) = 0$: The test is inconclusive and may represent an inflection point.
- Inflection Points: Points on the curve where the concavity changes sign ($f''(x) = 0$ with a sign transition).
Frequently Asked Questions (FAQ)
What does the constant of integration (+ C) mean?
When taking an indefinite integral, + C represents an arbitrary constant because the derivative of any constant is zero (d/dx [C] = 0). There are infinitely many antiderivative functions that differ only by a vertical shift.
Can a definite integral be negative?
Yes. If the function graph lies beneath the x-axis ($f(x) < 0$) over the interval $[a, b]$, the evaluated definite integral will yield a negative net signed area.
What is the geometric meaning of the first derivative?
The first derivative $f'(x_0)$ evaluates the exact numerical slope of the tangent line touching the function curve at point $x = x_0$, indicating the instantaneous rate of change.