C) DERIVADAS

Me parece que este video responde adecuadamente a la pregunta que nos hacemos :

¿Qué es el Cálculo ?







Les comparto el acceso a un buen libro de cálculo:
CÁLCULO DIFERENCIAL.




¿Qué es el cálculo?                     VEAN ESTE VIDEO. LO RECOMIENDO:





Notas sobre derivación ( Tecnológico de Costa Rica )
                              
https://docs.google.com/open?id=1pIQFC_J2CJSkwvs_fPE5-zHRuYRLynpzqlROf7AXfPLvCA7obGXqsf2wAPBA

Reglas de derivación


Sean a, b y k constantes (números reales) y consideremos a: u y v como funciones.

Derivada de una constante

Derivada de una constante

Derivada de x

Derivada de función identidad

Derivada de la función lineal

Derivada de función identidad

Derivada de una potencia

Derivada de una función potencial

Derivada de una raíz cuadrada

Derivada de una raíz cuadrada

Derivada de una raíz

Derivada de una función irracional

        






   


DERIVADAS ( VIDEO 2 )



VIDEO DERIVADAS COMO RAZÓN DE CAMBIO



DERIVACIÓN IMPLÍCITA


Funciones implícitas

Publicado por Profesor - 25/02/10 a las 02:02:58 pm

En unos días vamos a poner ejercicios de derivadas implícitas, pero antes, les dejamos por aquí la definición.

Funciones implícitas

En una correspondencia o también una función si está definida en forma implícita cuando no aparece despejada la y sino que la relación entre x e y viene dada por una ecuación la cual tiene de dos incógnitas cuyo segundo miembro es el cero.

Derivadas de funciones implícitas

Para poder  hallar la derivada correcta en forma implícita no es necesario despejar y. Así que basta el derivar miembro a miembro paso por paso, utilizando así todas las reglas vistas hasta ahora en derivadas.es  y teniendo presente lo siguiente:

x’=1.

En general y’≠1.

Por lo cual omitiremos x’ y dejaremos y’.

Derivación  implicita

Derivación implicita

Derivación  implicita

Derivación implicita

Y luego cuando las funciones son ya más complejas podemos utilizar una regla para facilitar el cálculo de la función:

Derivación implicita

Derivación implícita

Derivación implícita





DERIVACIÓN IMPLÍCITA




DERIVACIÓN SUCESIVA



Ejercicio de derivación sucesiva



1)   ¿Qué es la diferenciación ?
             http://mathforum.org/library/drmath/view/53399.html
     2)  Explica la Regla de la cadena en términos sencillos
         

PREGUNTAS Y RESPUESTAS FRECUENTES


Meaning of Derivative

Date: 10/12/96 at 18:6:19
From: Dominic Tsang
Subject: Meaning of Derivative

Can you give me a plain English meaning of the idea of a derivative? 

Thanks!

Date: 10/13/96 at 15:54:4
From: Doctor Scott
Subject: Re: Meaning of Derivative

Dominic,

The derivative does, in fact, have a nice "plain English" meaning. 
You probably learned that the derivative is the slope of the tangent 
line to the curve at a point.  The derivative describes the rate of 
change of the function at that point.  Actually, it is the 
*instantaneous* rate of change of the function at the point.  

For example, if you are blowing up a balloon, the volume of the 
balloon depends on the radius of the balloon.  That is, 
V = (4/3)*pi*r^3.  The derivative of V (with respect to r) would 
tell you how fast the volume is changing as the radius changes.

Hope this helps!

-Doctor Scott,  The Math Forum
 Check out our web site!  

Date: 10/13/96 at 16:2:48
From: Doctor Ken
Subject: Re: Derivative

Hi Dominic -

Here's an example of the derivative and what it means.  Let's say 
you've got a function, call it f(x).  The derivative of f, f'(x), 
tells you how fast f is changing.  If the derivative is positive, 
f is increasing, and if the derivative is negative, f is decreasing.  
If the derivative is 0, then at that point f is neither increasing 
nor decreasing.

Let's say f(x) represents the position of a car on a straight road at 
time x.  Then the derivative f'(x) tells you the velocity (that's 
like speed) of the car at time x.  If the car is going forward the 
velocity will be positive, and if it's going backward the velocity 
will be negative.

If you take another derivative, you'll call it f''(x), and that tells 
you the acceleration of the car at time x.  If the car is speeding up 
the acceleration will be positive, and if it's slowing down it will be 
negative.

Here's a little-known fact I'll let you in on: if you take another 
derivative, that has a name too.  It's called "jerk," and we write 
f'''(x). Can you figure out what the physical interpretation of jerk 
is?

The derivative is used all the time in physics, in exactly this kind 
of way.

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    

What is the Purpose of Determining a Derivative?

Date: 12 Jan 1995 12:57:41 -0500
From: Jenny Lay
Subject: Help!

Dear Dr. Math,

Earlier this semester we learned about derivatives in Calculus. I 
know how to determine the derivative of something, but what is the 
purpose?

Thanks.
Jenny

Date: 12 Jan 1995 14:11:43 -0500
From: Dr. Ken
Subject: Re: Help!

Hello there!

There are lots of reasons we'd want to take the derivative of
something.  First of all, let's say you're riding in your shiny 
new sports car and you have the best odometer in the world. It 
will tell you to the nearest thousandth of a mile (or something 
like that) how far you've gone. If you graphed what the odometer 
tells you as a function of time, so that time is on the x-axis and 
distance is on the y-axis, you could take the derivative of this 
function and figure out your speed for every point in your journey.  
So all the information about your speed and acceleration and
everything can be gotten from the odometer, as long as you know 
how to take derivatives.

Here's a question my calculus teacher once asked me: in cars, there's 
both an odometer and a speedometer. Essentially, the speedometer 
takes the derivative of the odometer information (before it gets to 
the odometer though; it's straight from the wheels). How does it do 
that?  It's been doing that since way before on-board computers 
happened to cars. So essentially, they've found a purely mechanical 
way to take derivatives. Neat stuff, worth researching.

The derivative is also quite an intuitive concept, I think.  Let's 
say you have a growth chart on your wall. If you're a human (which I 
believe you are) you'll probably have a couple of periods when you 
grew faster than at other times in your life. If the marks were made 
at regular intervals, they'd be more spread out in certain periods 
and more clustered together in others. So it's not hard to figure out 
from this chart that you grew faster in those growth spurt times than 
in the lull times. Well, how fast you grew is just the derivative 
with respect to time of how tall you were. So the derivative will be 
big sometimes, small sometimes, and once you hit 40 years old, it 
will be negative (some people say).

So these are a couple of real-life examples. Other examples that 
are based on integration (the inverse of differentiation) would 
include finding the volume of some objects, finding the area of 
some regions in a plane, and stuff like that. And trust me, if you 
go on and do some more in math, taking the derivative of functions 
will be SHEER BLISS compared with some of the more nasty stuff 
(which is more rewarding. Stick with math!).

So that's how I feel about derivatives.

Date: 12 Jan 1995 15:23:35 -0500
From: Dr. Elizabeth
Subject: Re: Help!

Hi Jenny!

One of the nicest things you can do with derivatives is to find out 
where the maximum value of a function is - which can be a very useful 
thing to know. The derivative of a function is its slope at any given 
point (actually it's the slope of the tangent line to the point, but 
even though a point can't have a slope, I always found it easier just 
to think of the derivative this way). At the highest point of a 
function (its maximum value), it's gone as far up as it's going to, 
and it's about to start heading back down. Its slope at this point, 
then, will be zero, since it isn't going up, and it isn't going down.  

Let's say you had a function and you wanted to know where, exactly, 
it would reach a maximum (I'll give you an example of why you might 
want to know this in a minute). Take the derivative of the function 
and set it equal to zero. Then solve for x or a or the number of 
feet of fence or the number of frogs or whatever it is that your 
independant variable happens to be.
  
Here's my example: I throw a ball straight up at a speed of 6 meters
per second somewhere where there's absolutely no air (no air 
resistance). When will it be at its highest point?

I can write the ball's height in an equation: the height at any 
time t will be the ball's upward velocity times the amount of time 
it's been going up (6m/sec times time, t) minus its gravitational 
acceleration downward times the amount of time it's been up there 
squared (10 m/sec/sec *that's the acceleration of gravity* times 
time squared, t^2).  

Leaving out the units so that the math is easier to see here:

   Height = 6t-10t^2.

If we take the derivative of this function, we have an equation for 
the slope of this function at any given time, t. The value of t for 
which this new equation is equal to zero is the same t at which the 
height of the ball will be a maximum.

Derivative of height = 6-20t = 0
                   6 = 20t
                   t = 6/20 second

The ball will reach its maximum height in 6/20 of a second.

Hope this helps! 
Elizabeth, a math doctor
    

Calculus Chain Rule

Date: 11/23/97 at 20:09:15
From: stuart
Subject: Calculus chain rule

I can't understand the chain rule. Every time I ask someone to 
explain it they use y's and u's, etc... could you give me the chain 
rule in easy terms, like how to do it, not just give me a formula 
like y=(U)^2?  
Thanks.  
Stu

Date: 11/23/97 at 21:24:09
From: Doctor Scott
Subject: Re: Calculus chain rule

Hi Stu!

Good question.  I was just skimming an excellent Calculus book written 
by Paul Foerster where this very question was addressed. His 
suggestion was that you should think of the chain rule as a process 
rather than a rule with a lot of du/dx and dy/dx's.  So, here goes....

Remember that the chain rule is used to find the derivative of 
*compositions of functions* - that is, functions that have functions 
inside of them. 

For example, the function sin(x^2) can be thought of as a composition 
of two other functions, sin x and x^2, with the x^2 being INSIDE the 
sin function. 

Similarly, the function (x^2 - 5x + 8)^(1/2) is also a composition of 
two other functions, (x^2 - 5x + 8) and x^(1/2), with the first 
function being INSIDE the second.  

One more example?  The function cos(tan(5x-3)) is the composition of 
three functions, 5x - 3 inside of tan x, inside of cos x.  

So the chain rule gets applied when there is some function INSIDE of 
another function.

The stuff that people have been telling you probably goes something 
like this: If y = sin(4x-3), then we can write this function as 
the composition of y = sin u and u = 4x - 3. (Again, notice that the 
4x - 3 is INSIDE of the sin function.)  Then, dy/dx = dy/du * du/dx.  
So, we have dy/dx = cos u * 4; but u = 4x-3, so we have dy/dx = 
4cos(4x-3).

How about another way? Let's think of the chain rule as a process. 
The derivative of a composite function is the DERIVATIVE OF THE 
OUTSIDE FUNCTION  TIMES  the DERIVATIVE OF THE INSIDE FUNCTION.

In practice, here's how it works. Consider y = sin(4x-3). The outside 
function is a sine function; its derivative is cosine, so we have (so 
far) cos(4x-3). Now, INSIDE the sine function is 4x-3. Its derivative 
is 4, so now we have 4cos(4x-3). Notice that there is no other 
function "inside" the 4x-3, so we are done.

Let's look at a couple more examples:

y = (x^2 - 5x + 8)^(1/2). The OUTSIDE FUNCTION is basically a power 
rule problem, so we have 0.5(x^2 - 5x + 8)^(-1/2) using the power 
rule. The INSIDE FUNCTION is x^2 - 5x + 8; its derivative is 2x - 5, 
so we have y' = (2x - 5)(.5)(x^2 - 5x + 8)^(-1/2).

y = cos(tan(5x-3)).  The outermost function is a cosine, so its 
derivative is negative sine: -sin(tan(5x-3)). Inside the cosine is a 
tan function; its derivative is sec^2, so we now have  

   sec^2 (5x-3) * (-sin(tan(5x-3))

Finally, inside of the tan function is 5x-3; its derivative is 5. 
So, FINALLY, we have 

   5 * sec^2 (5x-3) * (-sin(tan(5x-3))
 
Or, simplifying, we get  

   y' = -5 sec^2 (5x-3) sin(tan(5x-3))

So, it helps a lot to think of the chain rule as:  The derivative of 
the outside TIMES the derivative of what's inside!

-Doctor Scott,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/