We must acknowledge that so far we have always been working with the rate of change. By differentiating $y$ with respect to $x$, we get the derivative of $y$ with respect to $x$: $$ \begin{equation}\begin{aligned} \frac{d}{dx}(y)&=\frac{dy}{dx}\\ \end{aligned}\end{equation} $$
This is also referred to as the rate of change of $y$ with respect to $x$. Read the following rates of change:
- $\frac{du}{d\theta}$
- $\frac{dv}{dx}$
- $\frac{d\theta}{dx}$
- $\frac{du}{dt}$
- $\frac{ds}{dt}$
When the rate of change of a variable is with respect to time, we do not have to mention that it is with respect to time e.g. $\frac{du}{dt}$ can be simply read as the rate of change of $u$.
Significance of the rate of change when plotting graphs
When we use $\frac{dy}{dx}$, we have the rate of change of $y$ with respect to $x$. This is the gradient function when the graph of $y$ versus $x$ is plotted.
In like fashion, the derivative $\frac{du}{dt}$ is the rate of change of $u$ with respect to $t$ and represents the gradient function of the graph of $u$ plotted against $t$.
Challenge
Determine which quantity is plotted on the vertical and horizontal axes for the following derivatives to qualify as the gradient function:
- $\frac{du}{d\theta}$
- $\frac{dA}{dr}$
- $\frac{d\theta}{dx}$
- $\frac{ds}{dt}$
- $\frac{dv}{dx}$