Chapter 12.3, Problem 82E

To determine

To explain: the definition of the derivative of a function $f$ at a point $\left(x,f\left(x\right)\right)$ .

Answer to Problem 82E

Derivative of a function at a point $\left(x,f\left(x\right)\right)$ can be calculates using formula

  $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$ .

Explanation of Solution

Given information:

Given graph

   Calculation:

Consider the graph of function $f$ , secant line and tangent line:

In the left graph, the slope $\left(m\right)$ of the secant line passing through the point $\left(x,f\left(x\right)\right)$ and $\left(x+h,f\left(x+h\right)\right)$ can be found using slope formula as

  $\begin{array}{c}\text{Slope}\text{of}\text{secant}\text{line}=\frac{\text{Rise}}{\text{Run}}\\ =\frac{f\left(x+h\right)-f\left(x\right)}{h}\end{array}$

Now if $h$ will approach zero, the second point on curve $\left(x+h,f\left(x+h\right)\right)$ will come close and close to first point $\left(x,f\left(x\right)\right)$ . Thus when $h\to 0$ , secant line will become tangent line. Therefore

  $\begin{array}{c}\text{Slope}\text{of}\text{tangent}\text{line}=\underset{h\to 0}{\lim }\left(\text{Slope}\text{of}\text{secant}\text{line}\right)\\ =\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\end{array}$

Thus derivative of a function at a point $\left(x,f\left(x\right)\right)$ can be calculates using formula

  $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$