Derivative Calculator

The derivative of a function f(x) measures the instantaneous rate of change of f with respect to x. Geometrically, it's the slope of the tangent line at a given point. Algebraically, it's defined as the limit: **Definition**: f'(x) = lim(h→0) [f(x+h) - f(x)] / h For power functions, this limit yields a beautifully simple rule. **The Power Rule**: For f(x) = ax^n, the derivative is f'(x) = anx^(n-1) In plain language: bring the exponent down as a coefficient, then reduce the exponent by 1. **Proof Sketch**: Using the binomial theorem to expand (x+h)^n and taking the limit as h approaches 0, all terms with h² or higher vanish, leaving nx^(n-1). **Examples**: - f(x) = x² → f'(x) = 2x - f(x) = 3x⁵ → f'(x) = 15x⁴ - f(x) = 7x → f'(x) = 7 - f(x) = 4 → f'(x) = 0 (constants have zero slope) - f(x) = x^(1/2) → f'(x) = (1/2)x^(-1/2) = 1/(2√x) - f(x) = x^(-2) → f'(x) = -2x^(-3) = -2/x³ The power rule works for ALL real exponents: positive, negative, fractional, and even irrational. **Why It Matters**: 1. **Physics**: Position → velocity → acceleration are all derivatives. If position is s(t) = 4.9t² (falling object), velocity is s'(t) = 9.8t, and acceleration is s''(t) = 9.8 m/s² (gravitational constant). 2. **Economics**: If total cost is C(q) = 100q + 0.5q², then marginal cost (cost of producing one more unit) is C'(q) = 100 + q. 3. **Engineering**: Optimization problems — find where f'(x) = 0 to locate maxima, minima, and inflection points. 4. **Biology**: Population growth rates, reaction rates in chemistry, enzyme kinetics all use derivatives. **Other Differentiation Rules**: - **Sum rule**: (f + g)' = f' + g' - **Difference rule**: (f - g)' = f' - g' - **Constant multiple rule**: (cf)' = cf' - **Product rule**: (fg)' = f'g + fg' - **Quotient rule**: (f/g)' = (f'g - fg')/g² - **Chain rule**: [f(g(x))]' = f'(g(x)) × g'(x) **Higher-Order Derivatives**: You can differentiate repeatedly. The second derivative f''(x) measures how the slope is changing (concavity). The third derivative (f'''(x)) and beyond are used in Taylor series and advanced applications. **Famous Examples**: - **Area of a circle**: A(r) = πr². A'(r) = 2πr = circumference! The derivative of area is the boundary. - **Volume of a sphere**: V(r) = (4/3)πr³. V'(r) = 4πr² = surface area! Again, derivative of volume is the boundary. These are not coincidences — they reflect deep geometric truths about how volume/area grows as dimensions change.