Every time we want to find the derivative of an equation, we don’t have to use the definition of derivative. Instead, we can use general rules of differentiation that have been derived from the definition.
For example, it can be shown using the definition that the derivative of the function $y = x^n$ takes the form $\frac{dy}{dx} = n x^{n-1}$.
The function $y = x^2$ is in the format $y = x^n$ where $n=2$. Hence the derivative is,
\[
\begin{aligned}
\frac{dy}{dx} & = n x^{n-1} \\
& = 2 x^{2-1} \\
& = 2x.
\end{aligned}
\]
The table below shows some common functions and their derivatives.
| $y=f(x)$ | $\frac{dy}{dx}$ |
|---|---|
| $c$ | 0 |
| $x$ | 1 |
| $cx$ | $c$ |
| $x^n$ | $n x^{n-1}$ |
| $cx^n$ | $cn x^{n-1}$ |
| $e^x$ | $e^x$ |
| $e^cx$ | $ce^cx$ |
| $\ln(x)$ | $\frac{1}{x}$ |
| $\ln(f(x))$ | $\frac{1}{f(x)}\frac{d f(x)}{dx}$ |
As well as these rules for certain functions, there are also rules for functions which take particular forms. For example, a function which consists of two functions divided by each together,
$$f(x) = \frac{x^2}{e^x}.$$
For functions such as this we would use what is called the quotient rule. If a function is of the form:
$$y = \frac{f(x)}{g(x)}$$
using the quotient rule, the derivative of $y$ is:
$$\frac{dy}{dx} = \frac{f'(x) g(x)-f(x)g'(x)}{[g(x)]^2}$$
where $f'(x)$ is the derivative of $f(x)$ and $g'(x)$ is the derivative of $g(x)$.
Summary
We have spent some time understanding the concept of differentiation from first principles. In ecological and epidemiological modelling, we will use differentiation in the form of ordinary differential equations. Here we will specify the derivative of a function in terms of biological processes.
This means that as well as understanding how to obtain the derivative of a function, we also need to understand how to obtain a function from it’s derivative. We do this using integration.