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This may seem like an overly trivial question, but I've just recently become confused about Langrange's 'prime' notation for derivatives (for example $f'(x)$).

I know for sure that f'(x) = \frac{\delta f(x)}{\delta x}.

But suppose we replace x with an expression, like 2x+1. Do we write f'(x^2+1) = \frac{\delta f(x^2+1)}{\delta x} or f'(x^2+1) = \frac{\delta f(x^2+1)}{\delta (x^2+1)}?

Does putting the prime around the function instead of between its letter and parentheses make a difference? For example what does (f(x^2+1))' mean?

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    Cam, if you're going for compactness of notation, but the limited utility of primes is troubling you, one compact notation I've seen used a capital D: $D_x \sin\;x=\cos\;x$ for instance.2010-12-18

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f' is a function, so f'(2x + 1) denotes f' applied to $2x + 1$, or $\frac{df}{dx}(2x + 1)$. For example if $f = x^2$ then f' = 2x and f'(2x + 1) = 4x + 2.

(f(x^2 + 1))' is the derivative of the function $f(x^2 + 1)$, which is 2x f'(x^2 + 1) by the chain rule.

This question highlights a weakness of the ' notation, which is that it always comes with an implied variable with respect to which you're differentiating. If this variable is clear from context there's no problem, but sometimes it isn't.

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    Personally, I wouldn't write $f=x^2$. I'd write $f(x)=x^2$ instead, but maybe this has already been discussed elsewhere.2010-12-18
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Often people write something like (x^2)'=2x or (e^x)'=e^x, but as you noticed yourself, this is ambiguous, and I never use this notation. You can write $\frac{d}{dx}x^2=2x$ instead, i.e., you don't have to put the $x^2$ into the numerator. This way it looks much clearer, I think (though not shorter, unfortunately). So for me it would be \tfrac{d}{dx}f(x^2+1) = 2xf'(x^2+1). (However, \tfrac{df}{dx}(x^2+1) = f'(x^2+1).)