Let's say we have a function as follows:

f(x) = 3x^{2} - 2x + 7

The goal is to find f '(5x).

I try to calculate that in two different ways:

Method 1:

f '(x) = 6x - 2

f '(5x) = 6(5x) - 2

__f '(5x) = 30x - 2__

f '(5x) = 6(5x) - 2

Method 2:

f(x) = 3x^{2} - 2x + 7

f(5x) = 3(5x)^{2} - 2(5x) + 7

f(5x) = 75x^{2} - 10x + 7

__f '(5x) = 150x - 10__

f(5x) = 3(5x)

f(5x) = 75x

That is two completely different results for f '(5x) !!!!!!

How can that be?

If you solve a problem in two (or many) different ways, the result should be the same.

So, why are the results different in this case?

Which one is correct?

And what is wrong in the line of reasoning that leads to the other?

Please help.

Thanks. ]]>