What is a derivative of a function?

Question

What does it mean by getting a derivative of a function?And how is it different from getting a limit?

I know how to find the limit of a function but really do not understand what it is all about,everyone just gives me the 'approaching' idea.

I hope you understand what i am trying to say. Just give me a good understanding of the limit and the derivative.

Do you understand what a limit of a function is? – 5xum Apr 08 '14 at 07:54 — 5xum, Apr 08 '14 at 07:54
If you're saying about evaluating limit,then i do – Uzair Apr 08 '14 at 08:04 — Uzair, Apr 08 '14 at 08:04

score 2 · Accepted Answer · answered Apr 08 '14 at 07:59

2

A limit i.e. $lim_{x\to a}f(x)=f(a)$, is the idea of evaluating what happens when $x$ gets nearer and nearer to a certain value (namely $a$ here).

What we consider is what happens when we approach $a$ from the left and from the right, if we have a "well behaved" function, we expect that if we approach from the left or the right $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ ($x\to a^-,\,x\to a^+$), then the both of the "limits" should be the same, and also the limit should be the function evaluated at that point.

A derivative, for example $\frac{df(x)}{dx}$, is a measurement of what happens to our function as our variable changes, "$dx$" represents an infinitesimally small change in $x$, so the derivative measures a rate of change, be that space ($x$) or time ($t$).

answered Apr 08 '14 at 07:59

Ellya

11,783

is limits all about finding what happens to a functions at a point – Uzair Apr 08 '14 at 08:01
yes exactly that. – Ellya Apr 08 '14 at 08:01
for instance $f(x)=x$ is a "well behaved" function, since $lim_{x\to a^-}x=lim_{x\to a^+}x=lim_{x\to a}x=a=f(a)$.
But $f(x)=\frac{1}{x}$ is not a "well behaved" function at $0$ since:$,,,,,,$ $lim_{x\to 0^-}\frac{1}{x}=-\infty$ and $lim_{x\to 0^+}\frac{1}{x}=\infty$, and $f(0)$ is undefined.
– Ellya Apr 08 '14 at 08:03
if you could explain the concept of derivative(intuitive) clearly it would be much better ,for a beginner – Uzair Apr 08 '14 at 11:07
1

Okay so we'll think about a function of on variable, $y=f(x)$, intuitively we know that if $x$ changes, then $ y$ will most likely change as well (unless we have a constant function). But when we consider a derivative, we are considering how much $y$ changes when $x$ changes a tiny tiny bit (infinitesimally). This is why a derivative is a rate of change. – Ellya Apr 08 '14 at 13:07
For example, the derivative of a constant function $f(x)=c$ is $0$, because $f$ does mot change if $x$ changes, so the rate of change (ie the Derivative) is $0$. – Ellya Apr 08 '14 at 13:10

5xum · Answer 2 · 2014-04-08T08:21:28.617

I see two questions here.

First, let us adress the issue of limits. A limit of a function is defined, formally, like so:

$L$ is the limit of $f$ when $x$ approaches $x_0$, written as $$L=\lim_{x\to x_0}f(x),$$ if for every $\epsilon>0,$ there exists such a $\delta>0$ that for every $x, 0<|x-x_0|<\delta$, the inequality $|f(x)-L|<\epsilon$ holds.

The understanding of this is fairly simple: I look at the value of $f(x)$ around $x_0$. If all the values of $f(x)$ are close to $L$ if $x$ is close to $x_0$.

For continuous functions, the limit $\lim_{x\to x_0} f(x)$ will always equal $f(x_0)$. For example, the limit of $x$ as $x$ approaches $x_0$ is simply $x_0$.

The catch with limits is that the function $f(x)$ does not need to be defined at the point $x_0$. For example, I can define the function $f(x)=\frac{x}{x}$ which is not defined at the point $x=0$, since $\frac00$ is not defined. However, the limit of $f$ as $x\to0$ is simple, since $f(x)-1=0$ for every $x$ for which $0<|x|<1$, meaning that $\lim_{x\to 0} f(x) =1$.

The second question is about derivatives. Again, by definition, the derivative of $f$ in the point $x$ is defined as $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h},$$ if this limit exists. This definition comes from the fact that I want to draw a line through the point $(x, f(x))$ that is tangential to the graph of $f$. For this, I pick a value of $h$ and draw a line through two points, $(x,f(x))$ and $(x+h,f(x+h))$. I get some line which intersects $f$ in two points and which has a slope of $$\frac{f(x+h)-f(x)}{h}.$$ Now, I decrease the size of $h$. The smaller $h$ gets, the closer the line I draw gets to being tangential to the graph of $f$. The line I am approaching has a slope of $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h},$$ which is what I call a derivative of $f$ in the point $x$.

What is a derivative of a function?

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