11.1 Derivatives

In calculus, the slope of a function is a simplistic term for it’s derivative. Take for example the very general function \(f(X) = aX^b\). This function uses two parameters (\(a\) and \(b\)) and one variable (\(X\)) to return a number or function value \(f(X)\). If you think about it, this is exactly what the deterministic component of our regression does. When \(b\neq1\), this function is non-linear. Therefore, to determine the slope - the increase in the function value given a unit-increase in \(X\) - we need to take the derivative. The general formula for a derivative is given by

\[\frac{\Delta f(X)}{\Delta X}=abX^{b-1}\]

Note that we have used this derivative formula before, only we used it when \(f(X)=Y\) and the formula was linear (i.e. \(b=1\)) and \(a = \beta\).

\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\]

\[\frac{\Delta Y_i}{\Delta X_i}=\beta_1\]