# $\frac{d}{dt}|_{t=0}$ What does this notation mean? [closed]

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

What does this notation mean?

• It means $f'(0)$ Jul 27 at 19:10

It means $$f'(0)$$. More generally,$$\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=a}f(t)=f'(a).$$
• 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 $\displaystyle{\left.\frac{\mathrm d}{\mathrm dt}f(t)\right|_{t=a}}$?
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 $$\begin{equation*}\frac{dx^n}{dx}|_{x=2}=nx^{n-1}|_{x=2}=2^{n-1}n\end{equation*}$$
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