Differentiation & composite functions

[kingwinner]

What is the actual meaning of the notation of

df[g(x)]
-------
  dx

Here do we actually differentiate f with respect to x first and then evaluate it at g(x), or do we first evaluate f at g(x), then differentiate it with respect to x? Does the order matter?

df
--[g(x)]
dx

d
--f[g(x)]
dx

Are these two equivalent to the first one? (i.e. do all three notations have exactly the same meaning?)

Thanks for explaining!

 

[Ben Niehoff]

Have you learned the chain rule yet?

Let

f = f(g)
g = g(x)
h = f(g(x))

Then the chain rule is precisely the statement that

$$\frac{dh}{dx} = \left. \frac{df}{dg} \right|_{g(x)} \; \frac{dg}{dx}$$

In other words, the chain rule states that yes, your two methods described are equivalent.

 

[HallsofIvy]

What d f(g(x))/dx really means is that you take the composite function: first apply g to x then apply f to that. Whatever function you get is what you differentiate.

For example, if g(x)= 3x- 4 and f(x)= x², then f(g(x))= (3x-4)²= 9x²- 24x+ 16. The derivative of that is 18x- 24.

As has been pointed out, you could also use the "chain rule". Letting u= g(x), f(g(x)) becomes f(u) and the chain rule says df(g(x))/dx= (df(u)/du)(du/dx)= (df(u)/du)(dg/dx).
Since f(u)= u², df(u)/du= 2u and since g(x)= 3x- 4, dg/dx= 3.

(df(u)/du)(du/dx)= (2u)(3)= 6u= 6(3x-4)= 18x- 24 as before.

 

[kingwinner]

For example, if g(x)= 3x- 4 and f(x)= x², then f(g(x))= (3x-4)²= 9x²- 24x+ 16. The derivative of that is 18x- 24.

But the following notation seems to give inconsistent results:
df
--[g(x)]
dx

In your example,
df
--(x) = df/dx = 2x (df/dx is a function of x)
dx

df
--[g(x)] = 2(3x-4) = 6x-8 (evaluate the derivative function at g(x) )
dx
which is different from 18x - 24

So this is what I am concerning with, there are 3 notations (see top post) that seemingly represent the same thing, but can get different answers because their interpretations can be different...

Please help...this is what I've been wondering (and having trouble understanding) for a long time...and the 3 different ways of writing a derivative as shown in the first post come up very often.

 

[HallsofIvy]

But the following notation seems to give inconsistent results:
df
--[g(x)]
dx

Yes, that notation does. That's why I would never use such notation!

In your example,
df
--(x) = df/dx = 2x (df/dx is a function of x)
dx

df
--[g(x)] = 2(3x-4) = 6x-8 (evaluate the derivative function at g(x) )
dx
which is different from 18x - 24

And I did not write any of those things. That is why I do not use that notation.

So this is what I am concerning with, there are 3 notations (see top post) that seemingly represent the same thing, but can get different answers because their interpretations can be different...

Please help...this is what I've been wondering (and having trouble understanding) for a long time...and the 3 different ways of writing a derivative as shown in the first post come up very often.