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I have a doubt which confuses me a lot. If f(x) is a function in x then does f’(2x) mean $d f(2x)/d 2x $ or $d f(2x)/ d x $ as when we integrate it the result is f(2x)/2 and the same for $ f’(x^2)$. Is it $d f(x^2)/d x^2 $ or $d f(x^2)/d x $ and what do we get on integrating it?

Sorry for the very petty doubt but it really confuses me. Any help would be greatly appreciated ! Thanks in advance!

Nil
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$f'(2x)$ means you first take $f(y)$ and differentiate it, to get $f'(y)$, and once you have it, you plug in $2x$ as your input. Evaluation always comes last. $$ df/dx |_{2x}$$ might be another way to write it.

$f'(x^2)$ means to plug in $x^2$ as the input to $f'(x)$

I often find it helpful to rename expressions as new variables so I can see the main idea and stick to definitions as much as possible.

RobertTheTutor
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  • I understand your answer, but does anyone else find the notation $f^\prime(2x)$ confusing. To me, it might suggest that you evaluate $f$ at $2x$ and then take the derivative, which I know is wrong. I wish the notation would make it more obvious that the derivative is evaluated first. – Chris Mar 23 '24 at 16:13