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How can we use derivatives in our real lives, I know that we have a lot of formulas to find the derivatives of a function,but why do we need them ?

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Did you look at, say, Wikipedia: Applications of derivatives? –  Zev Chonoles Mar 7 '13 at 7:08
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Many people live quite satisfying lives without knowing the concept. But it is essential in many branches of science or engineering, and useful in a number of other fields. –  André Nicolas Mar 7 '13 at 7:11
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I need them to understand acceleration. Otherwise I would find it very disconcerting to ride in elevators and not have any clue as to what was going on. –  Trevor Wilson Mar 7 '13 at 7:20
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I use them to lose money quickly in the stock market. They are like iGadgets, they are not needed but they can be useful. –  copper.hat Mar 7 '13 at 7:23
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Speedometers are pretty useful to have around... –  Jair Taylor Mar 7 '13 at 8:16

6 Answers 6

Whenever you want to know how quickly some quantity is changing, you are basically talking about a derivative.

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this is acceleration not derivation, right ? –  mounaim Feb 13 at 16:25

Derivatives in physics

You can use derivatives a lot in Newtonian motion where the velocity is defined as the derivative of the position over time and the acceleration, the derivative of the velocity over time. So, to summarise: $$\vec v(t)=\frac{d}{dt}\vec{OM(t)}$$ $$\vec a(t)=\frac{d}{dt}\vec v(t)$$

An application for this will be something like this :

An object travels with the following equation in a Galilean frame of reference.$$\vec {OM}(t)=\begin{cases} x=1\\ y=t^2\\ \end{cases}$$ (this example is totally made up it does not correspond to free fall some something like that)

You can therefore evaluate the velocity and the acceleration of the movement. Here we go : $$\vec v(t)=\frac{d}{dt}\vec{OM}(t)=\begin{cases} x=\frac{d}{dt}1\\ y=\frac{d}{dt}t^2\\ \end{cases} = \begin{cases} x=0\\ y=2t \end{cases}$$

So the velocity at an instant t of this body will be the vector $(0,2t)$. You can also do same thing for the acceleration to obtain the vector $(0,2)$.

What you can do with this?

You can use these properties to study the movement, for example, of your car if you know it's velocity at any instant, wihch is something doable. Another implementation of this kind of physics in daily life is the accelerometer built in you iPhone. This device can find how fast and to which direction you rotate your phone. That's how you can turn your phone to take a landscape oriented photo.

Derivatives in chemistry

One use of derivatives in chemistry is when you want to find the concentration of an element in a product. Here is the principle :

Say you have an solution $S_0$ that you want to determinate it's concentration $C_0$ in an element and you know that this solution gives an equivalence with the solution $S_1$ with a concentration of $C_1$. You have to take a volume $V_0$ of $S_0$ and measure it's pH. Then you just put little by little of the solution $S_1$. You have to know how much you put and note the new pH as you put the solution. When you plot the graph $y=pH(V)$, you will notice a curve like the blue one. The red one is it's derivative.

enter image description here

Let $V_E$ be the volume for which the derivative is maximum. Then, you are able to use the relation $$C_0V_0=C_1V_E$$ to determine the unknown concentration. Be careful, the relation above is just true where all stoichiometric coefficients are "1"s. You can adapt the relation if they are not "1"s.

Derivatives in math

I don't know you background about derivatives but the most common use of the derivatives in maths is for studying function. The most important theorem for studying functions is :

If $f$ is a function defined on an interval I, and $f'$ it's derivative, then if $f'>0$ then $f$ is increasing, if $f'<0$ then $f$ is decreasing and if $f'=0$ then $f$ has a minimum or maximum at that point.

This very important theorem let us study variations of functions. What about in everyday's life? Here is a situation :

enter image description here The curve below has the equation $y=x^2-1$ and represent a road whereas the point $A(0;2)$ represent a wireless transmitter. You want to get at the closest point to the transmitter to get the best signal with staying on the road. We're looking to find that point of the road, closest to the transmitter.

If you know a bit of analytic geometry, you can find that the distance between the transmitter and the road is given by the function $d(x)=\sqrt{x^4-x^2+5}$. Let's study the variations of $d^2$ to get rid of the root (we can do this because the variations of $d$ and $d^2$ are the same). So $d^2\text{'}(x)=4x^3-2x=2x(x-\frac{\sqrt{2}}{2})(x+\frac{\sqrt{2}}{2})$. We have the roots of the derivative : $-\frac{\sqrt{2}}{2}, 0, \frac{\sqrt{2}}{2}$. And the derivative is positive, negative and positive again. Therefore, $x=-\frac{\sqrt{2}}{2}$ and $x=\frac{\sqrt{2}}{2}$ are minima of the distance between the transmitter and yourself. If we check $d(-\frac{\sqrt{2}}{2})$ and $d(\frac{\sqrt{2}}{2})$, we will see that these values are equals, so we can say that the closest positions to the transmitter are for the values of $x=\pm\frac{\sqrt{2}}{2}$.

These exemples are just basic utilities of derivatives. Derivatives can be used in more complicated domains like Taylor series. But this isn't the tool you'll use in everyday's life. Hope you find this useful :)

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+1, great answer for a bad question! –  JMCF125 Feb 5 at 11:55

When I was teaching Maths 101 to Economics majors, one of my favourite examples was instant inflation, which is the derivative with respect to time of the logarithm of the prices.

So you may say I was catching two birds (derivatives! logarithms!) with one stone.

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Third derivatives are used to synchronize stoplights on the road so that you sped as little time waiting as possible!

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tl;dr: Netlfix might be using it via Machine learning!

The derivative at a very informal (and intuitive) level basically express the instantaneous rate of change of any function. For example, if you were on a hill, the rate of change could express how fast you are falling or increasing and thus, can be useful for optimization problems (i.e. maybe if you follow the path where the rate of change indicates a decrease, you might get to the bottom, right? If you where trying to fin a local minimum for example). You are asking why they are useful and I hope to provide a cool example from my own field: Computer Science.

Machine Learning

Machine learning is an area of Artificial Intelligence where machines learn to recognize patterns from data. Say that you where a user in Netflix and you want to get good recommendations for the movies you want to watch. The (machine) computer might want to optimize some objective function (maybe it want's to optimize the the "closeness" of stars that you rate movies compared to what it would have recommended or some function that more or less captures what will make you happy as a user) so that it can make you as happy as it can. So it might try to optimize this objective function with respect to the data that it has (i.e. it might try to optimize this in relation to the movies you have already rated) and thus create its own decision function to predict rating for you. Since there is an optimization problem involved to obtain this goal, it might use derivatives!

[In reality I don't know what netflix actually does, but there are machine learning algorithms that use derivatives in the way I described as, "searching for a local minimum by following the current derivative". An example could be gradient descent. which is the algorithm I described]

My point is not really if Netflix is using it, but that it can be used for computers to learn and predict! If you are interested in stock prices, then maybe you can apply your Machine learning knowledge to that itself. But the point is that derivatives are used to solve optimization problems and a cool application in modern computing is Machine learning!!

Hopefully, this will give you a more "real world" relation of how derivatives are being used to make your life better! Hope this helps.

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Since Moray95 did such a nice job on science, I offer you
Derivatives in mudding through life:

If you understand what a derivative is, you can state certain things very clearly and succinctly as well as avoiding errors in your thinking.

For example, a word that is coming up a lot in the healthcare debate is "bending the curve". I'll bet most people don't know what that really means. In math terms it means that the derivative of the healthcare spending curve is decreasing. Of course, it's still positive (meaning that the spending is still going up).

Similar games are being played about "reducing our deficit". Now to you and me that might mean that the amount we owe is going down, but that is not what our fine politicians mean. They mean that the yearly deficity is going down. So our debt is not going up as fast as it was. Again, the derivative is decreasing and still positive.

No one seems to be looking at the derivative of the cumulative growth in inflation, which, of course, drives the cumulative deficit down.

Then we have income taxes. Tax revenue is the tax rate times the amount taxed. Some say we need to lower the rates; some say we need to raise them. I say there is at least one optimal rate, in the sense that it would produce the maximum revenue. To see this, we note that if the tax rate is 0% there will be no tax revenue; and that if it is 100% there will be no tax revenue, because everyone will stop working and go home.

So the potential revenue curve has value 0 at both ends. Because we know about derivatives, we know Rolle's theorem which says that if a non-constant function is zero at both ends of an interval, it must achieve a maximum or minimum inside that interval. In this case it would be a max. Of course this is oversimplified because our tax rates are tiered; but the fundamental situation is the same.

Climate change is full of derivatives. How fast is the polar ice melting? How much water is that dumping in the oceans and how fast are they rising? At what rate are we dumping CO$_2$ into the air? Is the derivative of the that curve positive? negative? decreasing? increasing? How does that translate into degrees of warming (one senses the chain rule lurking here).

Then there are the scams. I just got an offer to have $10,000 per month for my retirement, without lifting a finger. Sounds pretty good. The scheme doesn't seem very workable however. I could say the derivatives just don't rise that fast.

And the lies. For example an individual who told us all he had a PhD in physics. We were discussing how coffee cools off in a cup and I said the derivative of the temperature curve was proportional to the square of the difference in temperature between the coffee and the surrounding air. Mr. PhD in Physics was completely baffled by this statement. What do you think I concluded?

It is perfectly easy to describe a situation by saying the derivative is decreasing, and anyone who understands derivatives will know what you mean -- and also what you do not mean and didn't intend to imply. Anyone who doesn't will have to use lots of words to describe the same situation, and can easily be fooled into thinking the wrong thing. And there are many such situations.

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