Appendix A — Calculus
A.1 Derivatives
A.1.1 Definitions
The derivative of a function \(f(x)\) is defined as the limit of the difference quotient as the increment \(\Delta x\) approaches zero:
\[ \frac{df(x)}{dx} = f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} \]
This limit can be interpreted as the slope of the tangent line to the curve \(f(x)\) at the point \(x\). If the derivative is positive, the function is increasing at that point, if it is negative, the function is decreasing.
If \(f(x)\) represents a line, the derivative will be a constant independent of \(x\) (since the slope of a line is constant). However, if \(f(x)\) is non-linear, then \(f'(x)\) will change with \(x\). The rate at which \(f'(x)\) changes with \(x\) is given by the second derivative:
\[ \frac{d^2f(x)}{dx^2} = f''(x) = \lim_{\Delta x \to 0} \frac{f'(x+\Delta x) - f'(x)}{\Delta x} \]
Critical points (i.e., maxima, minima and saddle points) will always a have a first derivative of zero.
A.1.2 General rules
The derivative of a sum of functions is the sum of the derivatives of the functions:
\[ \frac{d}{dx} (f(x) + g(x)) = \frac{df(x)}{dx} + \frac{dg(x)}{dx} \]
The derivative of a constant times a function is the constant times the derivative of the function:
\[ \frac{d}{dx} (c \cdot f(x)) = c \cdot \frac{df(x)}{dx} \]
The derivative of a product of functions is given by the product rule:
\[ \frac{d}{dx} (f(x) \cdot g(x)) = f(x) \cdot \frac{dg(x)}{dx} + g(x) \cdot \frac{df(x)}{dx} \]
The derivative of a quotient of functions is given by the quotient rule:
\[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2} \]
The derivative of a power of a function is given by the power rule:
\[ \frac{d}{dx} (f(x)^n) = n \cdot f(x)^{n-1} \cdot f'(x) \]
The derivative of a composition of functions is given by the chain rule:
\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]
A.1.3 Rules for basic functions
The derivative of a constant:
\[ \frac{d}{dx} c = 0 \]
The derivative of a polynomial (\(x^n\)):
\[ \frac{d}{dx} x^n = n \cdot x^{n-1} \]
The derivative of an exponential (\(e^x\)):
\[ \frac{d}{dx} e^x = e^x \]
The derivative of the natural logarithm (\(\ln(x)\)):
\[ \frac{d}{dx} \ln(x) = \frac{1}{x} \]