Derivative Notation: Varieties & Prevalence

[AznBoi]

Where are there so many different ways of writing the derivative of the function? Which one is the most commonly used by college professors and that looks the coolest? Probably $$\frac{dy}{dx}$$ right??

 

[AznBoi]

Btw, I think the beginning of calculus is soo interesting. I think the slope of point of tangency is awsome. Using limits to find the exact slope and it is pretty simple if you think about how it works. Is calculus basically just all of this, finding "rates of change of functions." Will it become even more interesting later on??

 

[Moridin]

There are a few that could be good to know.

The one you posted is called Leibniz notation. There is also f'(x) and similar called Lagrange's notation. Operator notation is Df for the first derivative, D²f for the second and so on. Newton's notation is

From my experience, Operator notation is by far the least used. It does not hurt to be able to recognize the rest of them as well. More info:

http://web.mit.edu/wwmath/calculus/differentiation/notation.html

Yes, calculus will become more interesting later on.

 

[AznBoi]

I also find the derivative very perplexing. How do you use the equaiton derived from the limit of the secant slope to determine the slope of any point on a graph? I know that you just plug the x value of the points into the equation but.. you know what I mean? Just how does that work... The limit of a secant slope gives basically the slope of a point (x,f(x)) but it seems werid how you can use that same equation (the derivative right?) to find the slop of ANY point on the corresponding graph. Plus, isn't the slope of the secant line different for any two sets of points? For example in a parabola.. How is the equation universal for all of the points in the equation?

Is all of what I've said so far true? I just find it weird and astonishing.

 

[Gib Z]

Well the limit gives the slop at point (x, f(x) )...and for any point of the corresponding graph, those are the points...you keep it general to all values of x by keeping it as a variable, x.

 

[AznBoi]

Well the limit gives the slop at point (x, f(x) )...and for any point of the corresponding graph, those are the points...you keep it general to all values of x by keeping it as a variable, x.

So just using f(x) and x gives you a general equation for the entire graph? That's interesting I've never though about it that way before. Is that the only reason for being able to use it to find the slope at any point on the graph?

 

[Gib Z]

Yes. Say we made the limit expression less general. We didn't want the derivative at any point x, but say x=3.

Then the expression becomes
$$\lim_{h\to 0} \frac{ f( 3+h) - f(3)}{h}$$ which only becomes the derivative at 3.

Just keeping it general allows it to be used for all values of x.

Here's an example: A quadratic ax^2+bx+c is general, which 3x^2-3.34b + pi is a more specific one.

 

[AznBoi]

Yes. Say we made the limit expression less general. We didn't want the derivative at any point x, but say x=3.

Then the expression becomes
$$\lim_{h\to 0} \frac{ f( 3+h) - f(3)}{h}$$ which only becomes the derivative at 3.

Just keeping it general allows it to be used for all values of x.

Here's an example: A quadratic ax^2+bx+c is general, which 3x^2-3.34b + pi is a more specific one.

oh okay I see now, thanks! Also, do you need to use the limit process to find derivatives every time? Are derivatives basically general equations of the tangential slope for the function?

 

[Gib Z]

do you need to use the limit process to find derivatives every time?

Well there are short cut's and patterns obviously. Instead of using the limit process for f(x) = x^n, we could prove the power rule.

Or Instead of using the limit for g(x) = x^3 - 3x^2 we would prove that the derivative is the sum of the derivatives of the 2 functions x^3 and -3x^2.

We can build on old derivatives and use the chain rule, product rule and quotient rule as well, instead of starting from limits every time.

For your other question - Yes. You will notice at points where the function is not defined, neither is the derivative.

 

[AznBoi]

oh yeah my teacher said that we will be learning shortcuts at the end of the calculus course. Basically you have to do everything the hard way before you can learn the easy way. What if I learn all of the shortcuts first? Would it be bad for me?

 

[Gib Z]

It would be terribly bad for you. The "hard" way, the limit way, gives us an understanding of what the derivative is. That is because from that definition we can see that derivative is taken by finding the gradient between 2 points and limiting the difference of these points to zero. This is how we know it is the tangent at a point.

Shortcuts don't work without the full method.

 

[AznBoi]

Oh okay. Thanks for your help.

 

[Hurkyl]

oh yeah my teacher said that we will be learning shortcuts at the end of the calculus course. Basically you have to do everything the hard way before you can learn the easy way. What if I learn all of the shortcuts first? Would it be bad for me?

Your calculus class teaches you many facts about derivatives. All of those facts are useful.

In the "real world", the problem you are trying to solve is virtually never "What is the derivative of this function?" -- in the "real world", you are generally trying to solve a more interesting problem, directly relevant to whatever research or analysis or computation you are trying to do. In order to solve those questions, you need to be able to recognize when derivatives can be applied, how to formulate a problem to involve derivatives, what information those derivatives tell you, and how to compute or approximate those derivatives.

 

[quasar987]

listen hurkyl. he wise man.

 

[Gib Z]

Cave person talk similar us. We slow English...De reev a teev? What that?