Difference between f '(2x) and [f(2x)] ' ?

[kingwinner]

Homework Statement

What is the difference between f '(2x) and [f(2x)] ' ?

If we integrate, is there any difference between
∫ f '(2x) dx and ∫ [f(2x)] ' dx?

Homework Equations

N/A

The Attempt at a Solution

N/A

Can someone please explain? I would really appreciate!

 

[Office_Shredder]

[f(2x)]' is the derivative of the function f(2x)

f'(2x) is the derivative of the function f(x) applied at the point 2x. An example may help:

f(x) = sin(x). Then f'(x) = cos(x). So f'(2x) = cos(2x)

On the other hand, f(2x)= sin(2x) So [f(2x)]' = 2cos(2x).

 

[Bohrok]

How would you write that in Leibniz notation?

 

[Office_Shredder]

I've often seen it written as:

$$f'(2x) = \frac{df}{dx} \Big |_{2x}$$

The vertical bar meaning 'evaluated at'. So this is the derivative of f evaluated at the point 2x. For the other case

$$[f(2x)]' = \frac{d[f(2x)]}{dx}$$

 

[g_edgar]

So you wouldn't write
$$\frac{d\,f}{d\;2x}$$
??

 

[Avodyne]

I've often seen it written as:

$$f'(2x) = \frac{df}{dx} \Big |_{2x}$$

I don't agree. It seems to me that the prime on the left means "take the derivative with respect to the argument", and the argument is 2x; the d/dx on the right means "take the derivative with respect to x".

Ambiguity is removed via

$$f'(2x) = \frac{df(y)}{dy} \Big |_{y=2x}$$