Derivative (Differentiation)
Differentiation in calculus is the process of finding the derivative of a function. The derivative measures how the function's output value changes over time (if time is the input value).
Rules of Differentiation
- Power Rule: If \[ f(x) = x^n \], then \[ f'(x) = n \cdot x^{n-1} \]
- Constant Rule: If \[ f(x) = c \] (where \[ c \] is a constant), then \[ f'(x) = 0 \]
- Constant Multiple Rule: If \[ f(x) = c \cdot g(x) \], then \[ f'(x) = c \cdot g'(x) \]
- Sum Rule: If \[ f(x) = g(x) + h(x) \], then \[ f'(x) = g'(x) + h'(x) \]
- Difference Rule: If \[ f(x) = g(x) - h(x) \], then \[ f'(x) = g'(x) - h'(x) \]
- Product Rule: If \[ f(x) = g(x) \cdot h(x) \], then \[ f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) \]
- Quotient Rule: If \[ f(x) = \frac{g(x)}{h(x)} \], then \[ f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2} \]
- Chain Rule (for composite functions): If \[ f(x) = g(h(x)) \], then \[ f'(x) = g'(h(x)) \cdot h'(x) \]