## Notation **The Standard Derivative:** For a differentiable function $X(t)$, the derivative is given by: $ \frac{d}{dt}X(t)=X^\prime (t) $ This expression represents the *normalized rate of change* at which $X(t)$ changes with respect to $t$, i.e. the slope in the [[Rise Over Run Method]]. **Differential Form:** In differential notation, we emphasize the infinitesimal change rather than the ratio itself. We express this change in $X(t)$ as: $ dX(t) = X^\prime(t)\,dt $ Here, $X^\prime(t)$ is the rate (or slope) and $dt$ is a very small increment in time. Thus, the differential $dX(t)$ represents the actual change in $X(t)$ over the tiny time interval $dt$. > [!note:] > In essence, the derivative tells you how fast $X(t)$ is changing, while the differential tells you how much $X(t)$ changes over a specific (infinitesimal) interval. ## Chain Rule When we have a function $F$ that depends on $X(t)$, that is, $F = F(X(t))$, the classical [[Differentiation Rules#Chain Rule|Chain Rule]] tells us how to compute the derivative of such a composite function. $ \frac{d}{dt} F(X(t))=\frac{d}{dX}F(X(t))* \frac{d}{dt}X(t) $ In differential form: $ dF(X(t))=F^\prime(X(t))*dX(t)$ This equation means that the infinitesimal change in $F(X(t))$ is obtained by multiplying the derivative $F^\prime(X(t))$ by the infinitesimal change $dX(t)$.