When should I use dy/dx versus f'(x) for derivatives?

[dranseth]

Homework Statement

I cannot seem to find my textbook and we just started derivatives. Can anyone tell me when I would use the notation dy/dx as opposed to something like f'(x)??

Thanks!

 

[jhicks]

f'(x) is just function notation. If you're given an f(x), it's equally correct to use f'(x) or d(f(x))/dx, for example. In the case of f'(x), the derivative with respect to x is implied.

 

[dranseth]

So it wouldn't ever matter which one I use? I could use both?

 

[jhicks]

You should stick with one for consistency. There's nothing wrong with using y' over dy/dx, or f '(x) over d(f(x))/dx; Just be consistent.

 

[rock.freak667]

So it wouldn't ever matter which one I use? I could use both?

Not loosely in a sense.

For example, if you have

$y=x^2$
You would write $\frac{dy}{dx}$ or $y'$ = $2x$ and NOT $f'(x)=2x$

 

[nicksauce]

The dy/dx notation is nice when you're actually manipulating these as if they're actually numbers. It's a physicist's favorite trick (but a mathematician's worst nightmare). I for one am not a fan of f'(x), except when there are higher order derivatives involved.

 

[Schrodinger's Dog]

I'd get used to both forms, as they seem to turn up alternately, at least in my experience.

For example the quotient rule:

Leibniz notation:

$$\frac{d}{dx}\left (\frac{u}{v}\right ) = \frac{\frac{du}{dx}\cdot v-u\cdot\frac{dv}{dx}}{v^2}$$

Newtonian notation:

$$h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

or in shorthand notation:

$$h'=\frac {f'\cdot g-f\cdot g'}{g^2}$$

You might see something similar as well which is a Newtonian notation:

$$\dot{h}(x)=\dot{f}(x)+\dot{g}(x) \longrightarrow \ddot{h}(x)=\ddot{f}(x)+\ddot{g}(x)$$

or

$$h\,\dot{}\,(x)=f\,\dot{}\,(x)+g\,\dot{}\,(x) \longrightarrow h\,\ddot{}\,(x) = f\,\ddot{}\,(x) + g\,\ddot{}\,(x)$$

occasionally but these aren't used very often.

Which is the same as saying:

$$\frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx}\longrightarrow \frac{d^2}{dx^2}(u + v)=\frac{d^2u}{dx^2}+\frac{d^2v}{dx^2}$$

in Leibniz notation.

They look more complicated between forms sometimes, but once you get the idea they're pretty obvious.