The Definition of a Derivative

The graphical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of the derivative of a function. We will use this definition to calculate the derivatives of several functions and see that the results from the definition agree with our graphical understanding. We will also look at several different interpretations for the derivative, and derive a theorem which will allow us to easily and quickly determine the  derivative of any fixed power of x

In the last section we found the slope of the tangent line to the graph of the function f(x)=x^{2} at an arbitrary point (x, f(x)) by calculating the slope of the secant line through the points (x, f(x)) and (x+h, f(x+h)),


and then by taking the limit of  m_{sec}  as h approached 0 (Fig. 1).  That approach to calculating slopes of tangent lines is the definition of the derivative of a function.

Definition of the Derivative:

The derivative of a function \mathrm{f} is a new function, \mathbf{f} ' (pronounced "eff prime"),

whose value at x is f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} if the limit exists and is finite.

This is the definition of differential calculus, and you must know it and understand what it says. The rest of this chapter and all of Chapter 3 are built on this definition as is much of what appears in later chapters. It is remarkable that such a simple idea (the slope of a tangent line) and such a simple definition (for the derivative f ') will lead to so many important ideas and applications. 

Notation: There are three commonly used notations for the derivative of \mathbf{y}=\mathbf{f}(\mathbf{x}):

\mathbf{f}^{\prime}(\mathbf{x}) emphasizes that the derivative is a function related to \mathrm{f}

D(f) emphasizes that we perform an operation on f to get the derivative of f

\frac{d f}{d x} \quad emphasizes that the derivative is the limit of \frac{\Delta f}{\Delta x}=\frac{f(x+h)-f(x)}{h}.

We will use all three notations so you can get used to working with each of them. 

f'(x) represents the slope of the tangent line to the graph of \mathrm{y}=\mathrm{f}(\mathrm{x}) at the point (\mathrm{x}, \mathrm{f}(\mathrm{x})) or the instantaneous rate of change of the function \mathrm{f} at the point (\mathrm{x}, \mathrm{f}(\mathrm{x}) ).

If, in Fig. 2, we let x be the point a+h, then \mathrm{h}=\mathrm{x}-\mathrm{a}. As \mathrm{h} \rightarrow 0, we see that \mathrm{x} \rightarrow \mathrm{a} and \lim\limits_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}=\lim\limits_{x \rightarrow a} \frac{f(x)-f(a)}{x-a} so

\mathbf{f}^{\prime}(\mathbf{a})=\lim\limits_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}=\lim\limits_{x \rightarrow a} \frac{f(x)-f(a)}{x-a}.

We will use whichever of these two forms is more convenient algebraically.

Source: Dale Hoffman,
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