In our study of limits and rates of change, we saw that the slope of the tangent line to the graph of \( y = f(x) \) at the point \((a, f(a)) \) is calculated by finding the limit of the difference quotient, and this is defined as
\[ \displaystyle\lim_{h\rightarrow 0} \dfrac{\Delta y}{\Delta x} = {\displaystyle\lim_{h\rightarrow 0}} \ \dfrac{f(a+h) - f(a)}{h}\]
provided this limit exists.
The slope of the tangent line is used to find the instantaneous rate of change of \( y \) with respect to \( x \) at \( x = a \).
In calculus, this limit is called the derivative of \( f(x) \) at \( x = a \). The process of finding the derivative is called differentiation.
By definition, the derivative, \(f'(x)\), of function \(f(x)\) for any value \(x\) in the domain of \(f\) is thus
\[ f'(x) = \displaystyle\lim_{h\rightarrow 0} \dfrac{f(x+h) - f(x)}{h} \]
if the limit exists.
In function notation (Joseph-Louis Lagrange, 1736-1813), the derivative of the function \(f\) with respect to \(x\) is symbolized as \(f'(x)\) and pronounced “\(f\) prime of \(x\).”
In Leibniz notation (Gottfried Wilhelm Leibniz, 1684), the derivative of \(y(x)\) is symbolized as \(\dfrac{dy}{dx}\) and pronounced “dee \( y \) by dee \( x \).” We can also use the short form \(y'\), pronounced “\( y \) prime.”
The domain of the derivative function depends on whether the value of the limit exists for all values within the domain of the original function.
The domain of the derivative function may be smaller than the domain of the original function.