1. The Geometric Meaning of a Derivative
At its core, a derivative measures the instantaneous rate of change of a function. Geometrically, the derivative f'(x) or dy/dx represents the exact slope of the tangent line touching the curve at point x.
2. The Limit Definition
Isaac Newton and Gottfried Wilhelm Leibniz formulated the derivative as the limit of the secant line slope as the interval h approaches zero:
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
3. Essential Differentiation Rules
- Power Rule:
d/dx [xⁿ] = n · xⁿ⁻¹(e.g. d/dx [x³] = 3x²) - Constant Rule:
d/dx [c] = 0 - Constant Multiple Rule:
d/dx [c · f(x)] = c · f'(x) - Sum Rule:
d/dx [f(x) + g(x)] = f'(x) + g'(x) - Product Rule:
d/dx [u · v] = u'v + uv' - Quotient Rule:
d/dx [u / v] = (u'v − uv') / v²
4. Finding Local Maximums & Minimums
To locate peaks and troughs of a curve, set the first derivative equal to zero: f'(x) = 0. These points are called critical points. The second derivative test (f''(x)) determines concavity:
- If
f''(x) > 0, the curve is concave up (local minimum). - If
f''(x) < 0, the curve is concave down (local maximum).