1. confusion with differentiation

hi, im going over differentiation atm but reading from a book, and it states that v=ds/dt

now could someone explain what d stands for? is it delta or what :S

because theres another equation a=dv/dt = d(squared)s/dt(squared) so does d stand for something that represents a single value if its being squared or what?

2. Originally Posted by j55
hi, im going over differentiation atm but reading from a book, and it states that v=ds/dt

now could someone explain what d stands for? is it delta or what :S

because theres another equation a=dv/dt = d(squared)s/dt(squared) so does d stand for something that represents a single value if its being squared or what?

$\displaystyle \frac{dy}{dx}$ is a different way of saying $\displaystyle f'(x).$ It means the derivative of $\displaystyle y=f(x)$ with respect to $\displaystyle x.$

$\displaystyle f''(x)= \frac{d^2y}{dx^2}$ means the second derivative of $\displaystyle y=f(x)$ with respect to $\displaystyle x.$

3. Originally Posted by j55
hi, im going over differentiation atm but reading from a book, and it states that v=ds/dt

now could someone explain what d stands for? is it delta or what :S

because theres another equation a=dv/dt = d(squared)s/dt(squared) so does d stand for something that represents a single value if its being squared or what?

$\displaystyle \frac{d}{dx}$ is simply Leibniz notation for saying 'The derivative of ___ with respect to x." In your case, the dx is a dt, so you're differentiating with respect to the variable 't', which stands for time.

Usually in mathematics and entry-level physics, 'S' stands for the Position function. It's a mathematical function that describes an object location relative to a fixed point with respect to time (i.e. Time is on the x-axis).

$\displaystyle V=\frac{ds}{dt}$ simply means that the derivative of the Position function is 'V', the Velocity function. The rate of change of the position function, with respect to time, will tell us it's velocity. Easy to understand right?

$\displaystyle A=\frac{dV}{dt} = \frac{d^2S}{dt^2}$ simply means that Acceleration is the first derivative of the Velocity function and/or the second derivative of the Position function. Why is there a squared over the d and t in the second derivative, you ask? That simply means it's the second derivative of the position function.

$\displaystyle \frac{d}{dt}\cdot\frac{dS}{dt} = \frac{d^2S}{dt^2}$

Cheers!

4. In plain English...

Here is a quote from Calculus Made Easy by Silvanus P. Thompson & Marvin Gardner.

"$\displaystyle d$ merely means "a little bit of." Thus $\displaystyle dx$ means a little bit of $\displaystyle x$;or $\displaystyle du$ means a little bit of $\displaystyle u$. Ordinary mathematicians think it more polite to say "an element of," instead

of "a little bit of". Just as you please. But you will find that these little bits (or elements) may be considered infinitely small."

5. Originally Posted by dbakeg00
In plain English...

Here is a quote from Calculus Made Easy by Silvanus P. Thompson & Marvin Gardner.

"$\displaystyle d$ merely means "a little bit of." Thus $\displaystyle dx$ means a little bit of $\displaystyle x$;or $\displaystyle du$ means a little bit of $\displaystyle u$. Ordinary mathematicians think it more polite to say "an element of," instead

of "a little bit of". Just as you please. But you will find that these little bits (or elements) may be considered infinitely small."
When you said "in plain english" I though it wasn't going to be confusing.

6. Originally Posted by Anonymous1
When you said "in plain english" I though it wasn't going to be confusing.

LOOOL exactly what i though.

confusing as hell...

maybe calculus isn't meant to be understood?

7. Originally Posted by eddie2042
LOOOL exactly what i though.

confusing as hell...

maybe calculus isn't meant to be understood?
Calculus is actually rather intuitive. Like everything else it just takes a little practice. The derivative is related to the slope of a function, but it is a rate of change.

Stop for a second.

Think about what "rate of change" means just with your knowledge of the english language, post what you think, and we will work through it together.

[]EDIT[] Sorry this was aimed at j55

8. I guess we all learn in different ways...that little paragraph was pretty clear to me

9. re

im getting to grips with it, but naturally its still slightly confusing, especially when you mentioned about it being a little bit of, that makes me think of it as a little bit of this divided by a little bit of that, but im assuming thats not the case and theres no actual division included? its just a representation, am i correct?

and as to your post anonayous1, rate of change simply means how fast or slow something is changing, or is it alot more complicated.

thanks for your posts though, great help!

10. Originally Posted by j55
a little bit of this divided by a little bit of that
Haha!

Originally Posted by j55
and as to your post anonayous1, rate of change simply means how fast or slow something is changing, or is it alot more complicated.
Okay, so we are changing at some rate. What does this mean?

Say we have a function f(x), the derivate of function f'(x), describes the slope at a point. This enables us to "move along the function" f(x) because we can map out its slope as its inputs change

f(x)
|
|
| /*
|/__________x (Sorry when I extend the function further it comes out jumbled. You should draw it out)
0 1..

This is some function f(x). say we take the derivative and obtain f'(x). Then f'(1) describes the slope at *. Or, the rate f(x) is changing when x=1.

So we are answering the question: at what rate does f(x) change when its inputs change?

Is this clear?

11. re

why the 'haha!' ?

and yeah it does make sense but i am still reading up on it when i can. The 1 above the f is that the same as a suffix? to represent the first value.

also i dont see how the actual question is answered. how do you show the rate at which f(x) changes?

and finally ' we are changing at some rate. What does this mean? ' i dont know what this means

12. Originally Posted by j55
why the 'haha!' ?
Because I thought $\displaystyle \frac{little-bit-of-this}{little-bit-of-that}$ was funny of course!

Originally Posted by j55
and yeah it does make sense but i am still reading up on it when i can. The 1 above the f is that the same as a suffix? to represent the first value.
The 1 is the point x=1 on the x-axis, the * is the the location of the function when x=1, (1,f(1)). You know like (x,y)...

Originally Posted by j55
also i dont see how the actual question is answered. how do you show the rate at which f(x) changes?
For nonlinear functions the rate at which f(x) changes depends on the point you are referring to. Mapping these instantaneous rates of change is what allows us to move along the function. The derivative is the tool that accomplishes this. For instance f'(1)= (a number) is the rate at which we are changing on the function at (1,f(1)).

Originally Posted by j55
and finally ' we are changing at some rate. What does this mean? ' i dont know what this means
For instance f'(1)= (a number) is the rate at which we are changing on the function at (1,f(1)).

13. re

right its really coming together now, might sound naive but the ds/dt isnt an actual equation but a notation showing derivative of what with respect to what? :S .

anyone?