#math/calculus If $y=f(x)$, the derivative can be written $ f'(x)=\frac{dy}{dx}=\frac{df(x)}{dx}=\frac{d}{dx}f(x). $ At a particular point $x_{0}$, write $f'(x_{0})$, $\left.\dfrac{dy}{dx}\right|_{x=x_{0}}$, or $\left.\dfrac{df(x)}{dx}\right|_{x=x_{0}}$. The notation $dy/dx$ emphasizes the instantaneous rate of change of $y$ with respect to $x$. Its units are “units of $y$ per unit of $x$.” It behaves suggestively like a fraction in some formulas, but it denotes a single limit and is not ordinary division. The differential-operator notation $\dfrac{d}{dx}$ means “differentiate with respect to $x$.” Repeated differentiation gives $f''(x)=\dfrac{d^{2}y}{dx^{2}}$; see [[2A2 - Second Derivatives]]. > [!Practice] > Marsden & Weinstein, p. 75, exercises 33–36.