What is derivative of a function PART 1

In a real function y = f(x), we want to understand how it changes with respect to x. What happens in a given interval I? Does $y$ increase? Does it decrease? How fast does it change?

For example, consider the following cases:

• In $I = [1,4]$, $\mathrm{y\; increases\; by\; 6.}$
• In $I=[1,3]$, $\mathrm{y\; does\; not\; change.}$
• In $I = [1,2]$, $\mathrm{y\; also\; remains\; constant.}$

How can we define a tool to measure this change in any interval? One way is to compare the change in $\mathrm{y\; (}$$\Delta y$) over an interval $\mathrm{I\; with\; the\; length\; of\; that\; interval\; (}$$\Delta x$). We express this as:

$$\frac{\mathrm{\Delta }y}{\mathrm{\Delta }\mathrm{x}}$$

If the change in $\mathrm{y\; is\; small\; compared\; to }$$\mathrm{\Delta }\mathrm{<span\; class="mord\; mathnormal">x,\; then </span>}$$\frac{\Delta y}{\Delta x}$ is close to 0. If $\Delta y$ is large relative to $\mathrm{\Delta }\mathrm{<span\; class="mord\; mathnormal">x,\; then </span>}$$\frac{\Delta y}{\Delta x}$ is also large.

We can also compute this ratio for very small intervals $\mathrm{\Delta }\mathrm{<span\; class="mord\; mathnormal">x,\; making </span>}$$\Delta x$ as small as we want. Under certain conditions, we can even calculate $\frac{\Delta y}{\Delta x}$ as $\Delta x$ approaches 0. In this case, $\Delta x$ gets closer to a specific point on the $x$-axis (which we denote as $c$), and we replace $\frac{\Delta y}{\Delta x}$ with:

$$\frac{dy}{dx}\text{or}{f}^{\mathrm{\prime }}(c)$$

This is called the derivative of $y$ at $\mathrm{c.}$

used concepts and vocabulary:

• the real function
• how it changes with respect to x.
• in a given interval 
•  Does $y$ increase / decrease?
• For example
• How can we define a tool to measure this?
• How can we measure this 
• to compare the change 
• is close to 0
• Under certain conditions
• something is approaches 0
• In this case