given d/dx[f(2x)]=f'(x) and f'(1)=1 find f'(2)
the choices are 1/4, 1/2, 3/4, 3/2, or 5/2 if it helpes
To find f'(2), we can use the chain rule and the given equation d/dx[f(2x)] = f'(x).
Let's start by differentiating d/dx[f(2x)] with respect to x. By applying the chain rule, we get:
d/dx[f(2x)] = f'(2x) * d/dx(2x).
Now, since d/dx(2x) is simply 2, the above equation simplifies to:
d/dx[f(2x)] = 2f'(2x).
We also know from the given information that f'(1) = 1.
Substitute x = 1 into the derived equation to get:
d/dx[f(2)] = 2f'(2).
Since f'(1) = 1, we can conclude that 2f'(1) = 2.
Therefore, f'(2) = 2/2 = 1.
So, f'(2) = 1.
Based on the given choices, the answer is 1/2.