Mathematics Calculus Derivative - Formal Definition | Limit | Continuity and...

Derivative – Formal Definition | Limit | Continuity and Differentiability

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In simple sentence, derivative is the slope of a tangent line. Slope means change in y direction over change in x direction $(\frac{Δy}{Δx})$. However, in this article we will try to understand the in fundamental concept of derivative in calculus. But to understand derivative we also need to have a basic understanding on what is limit. So we will start our discussion on formal definition of derivative with the basic concept of limit.

Limit of a Function:

Say, a function $f(x) = \frac{x^{2}-4}{x-2}$. In this function, we get an output for every possible input of x except +2. If we substitute x=2, then the function become $f(2)=\frac{0}{0}$, which is an indeterminate form. So we can say that at the point, $x=2$ we can not define the function. And therefore this brings the concept of limit. Because we are unable of determining the value of the function when $x=2$, therefore we can determine the value of the function when x is almost 2, or the value of x near about 2, or approaching to 2 from both left-side and right-side of 2.

Therefore, if x approaches to 2 from its left-side of number line, the function tends to become: x=1.9, f(1.9)= 3.9. when x=1.99, f(x)=3.99. Similarly if x approaches to 2 from the right-side of the number line, the value of the function tends to become: x=2.1, f(x)=4.1, when x=2.11, f(x)=4.11 and so on. Here we can can see that as x approaches to 2, f(x) approaches to 4. So, we can say, f(x)→4 as x→2. This is the concept of limit.

Definition: A function f(x) is defined when x is near the number a, but not a. It means  f has a value on some open interval that contains a, except possibly at a itself. Then we write:

$$\lim_{x\to a}f(x)=L,\quad\text{f(x)→L, as x→a.}$$

So, this means that we can get f(x) “as close as we want” to L by making x sufficiently close to a. Here L is the limit of the function. Now we can find out the value of function $f(x)=\frac{x^{2}-4}{x-2}$ when x→2 using limit:

$$\lim_{x\to 2}f(x)=\frac{(x+2)(x-2)}{(x-2)}\\
\lim_{x\to 2}f(x)=(x-2)=2+2=4,\\
\quad\text{bcoz, x→2, but x≠2}$$

When does the Limit Exist: limit of a function f(x) exists, only when for some value of  x=a, $f(a)=\frac{0}{0}$. Also if for any value x=a, $f(a)≠\frac{0}{0}$, then we can say that the limit does not exists. For example, for the function $f(x)=\frac{|x|}{x},$ limit does not exists. before going to definition of derivative, let get to know a little about continuity.

Continuity:

A function is said to be continuous if we able to graph the function without taking off our pen, or pencil. So the definition of continuity refers that A function is continuous at a point x=a, if 

$$\lim_{x\to a} f(x)= f(a)$$

as x approaches to a, the y value of x, f(x) will be equal to y value of a, f(a). Which means at the point a, f(x)=f(a), when x→a. And to find f(x) = f(a), we have to approach x→a from both left-side of a, x→$a^{-}$ and right-side of a, x→$a^{+}$. Therefore,we can define that a function is continuous if:

$$\lim_{x\to a^{-}} f(x)=\lim_{x\to a^{+}} f(x)=f(a)$$

The formal definition of Derivative

At the beginning we have said that derivative is the slope of tangent line. Now we will see what that really means. We know, a slope (m) is the rate of change of vertical variable (y), with respect to horizontal variable (x). So, $m=\frac{Δy}{Δx}$.

Derivative | The Formal Definition | Limit Of Function | Continuity

This is how we can find the slope for straight line. Change of y variable, $Δy=y_{2}-y_{1}$, with respect to change of x variable $Δx=x_{2}-x_{1}$. But for any curve y=f(x), slope just touches its surface. Let slope touches the curve at the point $p(x, f(x))$. Therefore, to find slope at p, we have to imagine another point $p_{1}((x+h), f(x+h)),$ closer to p, such that h→0, but h≠0. Because the nearer the point $p_{1}$ to p, the accurate the value of slope will be. So in this case the slope of the curve at point p:

$$Slope=\frac{f(x+h)-f(x)}{(x+h)-x}$$

Derivative | The Formal Definition | Limit Of Function | Continuity

But when h=0, the above slope become the form $\frac{0}{0}$, So h≠0 but h→0, therefore we have to include limit to define slope of the curve:

$$\lim_{h\to 0}\frac{f(x+h)-f(x)}{(x+h)-x}$$

This is called the derivative of the function f(x) at the point $p(x, f(x))$. So,

$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{(x+h)-x}$$

The line that touches the curve y=f(x) at p, refers to as tangent line. As above slope we derive from the tangent line, therefore the derivative of a function defined as slope of tangent line. And this definition of derivative also refers to as formal definition of derivative.

Derivative of a function y originally expressed as $\frac{dy}{dx}$. because it is similar to $\frac{Δy}{Δx}$, but in this case Δx→0, also Δy→0.

Which means, $\frac{dy}{dx}$ indicates the instantaneous change of the function, or infinitesimal change in y with respect to infinitesimal change in x. Also indicate how fast that change happen.

Differentiability

if (x+h) indicates a point c on x-axis, then (x+h)=c, and y=f(x+h)=f(c), such that x→c. Then we can define the derivative of the function f(x) at the point x=c:

$$f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$$

Here, we can define that if f is differentiable at x=c, then f is continuous at x=c. So if we know that f is differentiable then we can find the limit at x=c exists, and also the function is continuous at x=c. We also can say that if f is not continuous at x=c, then f is not differentiable at x=c. A function can be continuous but not differentiable, see here. Therefore we can say that a function to be differentiable at the point x=c, its limit must exist at that point. know more.

Read more article on calculus, here

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