How to do derivatives

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

\vec$$ $$\vec a(t)=\frac
\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.$$\vec (t)=\begin x=1\\ y=t^2\\ \end$$ (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

\vec(t)=\begin x=\frac
1\\ y=\frac
t^2\\ \end = \begin x=0\\ y=2t \end$$

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.

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.

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 $d(x)=\sqrt$. 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 :) how to do derivatives


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