Find d/dx[f(x)] if d/dx[f(3x)]=6x

Question

I found the derivative of f(3x) as f'(3x) * 3

I then got f'(3x)=2x
Idk what to do now

Answer 1

Hi Jeffrey,

 

An interesting question.  Let's do a substitution.  Let u = 3x.  Then du/dx = 3.  By the chain rule,

6x = 2u

6x =  d    f(3x)

        dx

 

    = d   f(u)

       dx

 

     =  du    d  f(u)
         dx    du

 

     =  3    d   f(u)

               du

     

 

So we have

 

2u = 3 d f(u)
          du

 

 2 u = d   f(u)

3         du

 

But the variable name u doesn't matter, so it is just as true that

 

d  f(x) = 2 x

dx          3

 

 

Now, let's check.  Let's guess that f(x) = 1/3 x²

Then d/dx f(x) = 2/3 x, and d/dx f(3x) = d/dx 1/3 (3x)² = d/dx 1/3 9 x² = d/dx 3x² = 6x.  √

 

I hope that helps.

 

Regards,

Mike N.