What is derivative of a function PART 3 (y=x^2)

Let's try to understand derivatives a bit more through their graphical interpretation. In the first part, we saw how we can represent changes in a function using a straight line over a specific interval. We express this change as the difference in $y$ divided by the difference in $x$, or $\Delta y / \Delta x$

We can also examine what happens to $\Delta y / \Delta x$ as the interval shrinks toward a specific point. We can do this manually by moving the value of $b$ closer to $a$. In this case, we are talking about the derivative of $y$ at point $a$

Another important observation is a key property of the straight line that represents these changes. In the limit, as $b$ approaches $a$, this line has only one intersection point with $f(x)$.

Note: In the limit, this line has a specific name: the tangent line.

Not all functions have this property, but when they do, we can use it as an alternative definition of the derivative. We can find the tangent line and then associate its slope with the derivative of the function.

Now, let's apply this concept:

Find the intersection between $y = x^2$ and a straight line $y = mx + n$.

We set their equations equal to each other:

$$x^2 = mx + n$$

which gives us the quadratic equation:

$$x^2 - mx - n = 0$$

We cannot solve this equation directly because we do not have values for $m$ and $n$. However, we know that the tangent line must have only one point of intersection with $y = x^2$, which means the quadratic equation must have exactly one solution.

For a quadratic equation , a unique solution occurs when the discriminant is zero:

$$m^2+4n=\mathrm{0}$$

In this case, the solution for $\mathrm{x\; is:}$

$$x = \frac{m}{2}$$

This means that the slope of the tangent line at any point $x$ is $m = 2x$

So, for $f(x) = x^2$, the derivative is: f '(x) = 2x

vocabulary used:

Let's try to understand a bit more
through the graphical interpretation
a straight line
the interval shrinks toward a specific point.
to apply this concept
the discriminant is zero

$$\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$$

What is derivative of a function PART 2 (simplest example)

In the last post, we explored the graphical definition of the derivative. Now, we can use this idea to solve a simple problem.

Can we calculate the derivative of a constant function at every point? Let’s consider the function $y = c$, where $c$ is a real constant.

Yes, we can. We now understand that the derivative measures how a function changes over an interval as that interval approaches a specific point. However, a constant function does not change at all. Therefore, the derivative of any constant function is 0 at every point $x$.

new terms and vocabulary

• graphical definition
• solve a simple problem
• where $c$ is a real constant
• how a function changes over an interval
• Therefore
• does not change at all.

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