# Why do we take a derivative?

I don't have a fundamental understanding of what taking a derivative means and I don't understand why would we do it. I just know how to mechanically do it.

-
There is a set of videos on YouTube named "The big picture of Calculus" by MIT Opencourse ware. You should watch them, really helpful. –  Haider Rehman Butt Apr 26 at 18:16

If $f(x)$ is a (real) function (of a real variable), then its derivative $f'(a)$ at a point $a$ measures the sensitivity of $f$ to small changes in $x$ around $a$. For example, suppose $x$ is "the height of the sun in the sky" and $f(x)$ is "the length of the shadow cast by this tree." Then $f'(a)$ is a measure of how sensitive the length of the shadow is to a small change in the height of the sun if the height is around $a$. Experimentally, $f'(a)$ will be small if the sun is high in the sky ($a$ is large) while $f'(a)$ will get larger as the sun gets lower in the sky ($a$ is small). The derivative gives us a way to quantify this observation.

I decided not to use the term "rate of change" because I think some students have heard this term so many times that they have become immune to it, so perhaps the above will make more sense.

-
Very nice explanation. I may have to start using sensitivity to explain derivatives from now on. +1 –  Jack Henahan Jun 17 '11 at 22:45
I got the term from Deane Yang (I think) over at MathOverflow. –  Qiaochu Yuan Jun 17 '11 at 23:21
+1. This explanation makes me smile! :] –  Lyrebird Jun 18 '11 at 1:46
@Qiaochu Yuan +1. I like very much your answer. Referring to "small changes in $x$ around $a$", how can I define small? Small related to $a$? –  Alessandro Jacopson Jul 6 '11 at 16:20
@uvts_cvs: well, small really means "infinitely small." In practice I guess it means "small enough so that $f$ does not change much." –  Qiaochu Yuan Jul 6 '11 at 16:36

It's a pretty good, non technical explanation of the derivative and why one might care about it.

-

The derivative can be interpreted a number of ways, depending on context. For a simple physical perspective, you can think of it as a rate of change. If you want to know the rate of change at some given point (say, velocity 3 seconds into the movement of a thrown ball), taking the derivative will give you your answer. From a geometric perspective, taking the derivative tells you the slope of the tangent line at a given point.

From a historical perspective, this is sort of a workaround for Zeno's paradoxes of motion.

-

I don't know what kind of explanation you have gotten before, if any, so I think that I will start with one of the usuals. Let me first introduce to you the difference quotient: $$\frac{f(x+h)-f(x)}{h}$$ It has a very nice geometrical interpretation.

In this picture, the curve is the graph of a function. I hope that you can understand from my picture that the length of the horizontal red line is $h$, while $f(x+h)-f(x)$ is the length of the vertical red line. So $f(t)$ increases by an amount of $f(x+h)-f(x)$ when $t$ increases by an amount of $h$ from $x$. This means that the average increase on the vertical axis per unit of the horizontal axis is given by the difference quotient $$\frac{f(x+h)-f(x)}{h}.$$ So the difference quotient is the average increase of a $f(t)$ on the interval $[x,x+h]$. Another way to put it is that the difference is the slope of the line passing through the points $(x,f(x))$ and $(x+h,f(x+h))$, like this:

So the green line $g(t)$ can be written as $$g(t) = \frac{f(x+h)-f(x)}{h} t + b$$ where $b=g(0)$ (the point where $g$ crosses the vertical axis).

When we take the derivative of $f$ at $x$ we find the limit of the difference quotient as $h$ goes to $0$, or $$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}.$$ This is the definition of the derivative. Imagine that $h$ gets smaller and smaller. Then the point $(x+h,f(x+h)-f(x))$ will get closer and closer to the point $(x, f(x))$, and the green line will get closer and closer to the tangent line of $f$ at $x$.

So when $h \to 0$, the difference quotient becomes the slope of the tangentline, and now we have an interpretation of the derivative:

The derivative $f'(x)$ is the is the slope of the tangent line of $f$ at $x$.

Let's finish off by trying to find the derivative of the function $f(x)=x^2$ at the point $x=1$ only with the help of the limit stated above. $$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h\to 0} \frac{(x+h)^2-x^2}{h}$$ $$= \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h} = \lim_{h\to 0} \frac{2xh+h^2}{h}$$ $$= \lim_{h\to 0} 2x+h = \underline{2x}$$ When we plug in $x=1$ we get that $f'(1) = 2$. Hope this helps!

-

A derivative can be interpreted as the slope of a tangent line or a rate of change (in particular, a velocity). You might e.g. look at these notes

-

Derivatives are useful for optimization; finding the extrema (maximum or minimum) of functions. This is a very common task in applied mathematics.

-

-

if you given a function and you asked to find the behave of the given function then you may take the derivative to the given function, but if you given a behave of any function you need to take the integral of the behave function, to find the function of the given behave.

-
That's not easy to understand. Are you sure you are answering the right question? Or answering anything at all? –  Daniel R Jun 9 '14 at 20:20

The example that helped me understand goes like this (just to build intuition): Let's say your spending is recorded for a year, right.... jan 01- 100, feb01- 150, march 01- 80 - (but continuously, every week, day, hour, minute....)etc. and a graph and a function is now known - you can plot your spendings in relation to time. Just do the plot of your spending.

Now the most often question: how much did i spend ( say on march 01)? a: you plug number x to you function (3 in this example) and get the the result: i spent 80. And you can do that with every time value. so you know how much you spent at anytime.

The other question is: was I saving or spending lavishly in certain moment?. Say March 01. And that somehow requires comparison in two values. And comparison intuition, in my opinion, is the key.

Now, you can try answer it: say in feb 01= 100 and in march 01 =80. that means you were saving? Right?. but if you plug in feb 27 (let say you spent 79 on that day) so ,in comparison, on march 01= 80 you were spending lavishly.and you can take any time value you wish. the question is still relevant.

So derivative helps to answer the question with any given time. Was I spending or saving (and how good was i spending or saving)?.And be precise with the answer

-