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$\frac{d}{dt}|_{t=0} f(t)$

What does this notation mean?

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    $\begingroup$ It means $f'(0)$ $\endgroup$
    – saulspatz
    Jul 27 at 19:10
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It means $f'(0)$. More generally,$$\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=a}f(t)=f'(a).$$

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    $\begingroup$ Hi José. In your experience, is it more common to write it as $\displaystyle{\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=a}}f(t)$ or $\displaystyle{\left.\frac{\mathrm d}{\mathrm dt}f(t)\right|_{t=a}}$? $\endgroup$
    – Joe
    Jul 27 at 19:27
  • $\begingroup$ @joe The second option, by far. $\endgroup$ Jul 27 at 19:32
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If $g(x)=\frac{d}{dx}f(x)$, then $\frac{d}{dt}|_{t=k}=g(k)$. In other words, it is the slope of $f(x)$ at point $P=(k, f(k))$. As an example take the function $f(x)=x^n$ and say you want to find the slope at the point where $x=2$. Then you write \begin{equation*}\frac{dx^n}{dx}|_{x=2}=nx^{n-1}|_{x=2}=2^{n-1}n\end{equation*}

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generally you would put it after the function being differentiated like: $$\left.\frac{d}{dx}f(x)\right|_{x=a}\equiv f'(a)\tag{1}$$ the reason people often write it like this is to avoid ambiguity, as if you wrote it as: $$\frac d{dx}f(a)= 0$$ as $f(a)$ is a constant. But note that both sides in eqn. $1$ are considered the same just different notation

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