Consider the case of finding the average rate of change between two points on the Cartesian plane:
$$ \begin{equation}\begin{aligned} m&=\frac{\Delta y}{\Delta x}\\ &=\frac{y_2-y_1}{x_2-x_1}\\ \end{aligned}\end{equation} $$If we bring the two points infinitesimally close to each other, we get the notation: $$ \begin{equation}\begin{aligned} m&=\frac{dy}{dx}\\ \end{aligned}\end{equation} $$
This is the instantaneous rate of change a.k.a. the derivative of $y$ with respect to $x$. This is the gradient function of the graph of $y$ plotted against $x$.
Power rule
This rule is used to differentiate algebraic terms. Given the function, of $y$ in terms of $x$: $$ \begin{equation}\begin{aligned} y=x^n\\ \end{aligned}\end{equation} $$
The derivative is: $$ \begin{equation}\begin{aligned} \frac{dy}{dx} &= nx^{n-1}\\ \end{aligned}\end{equation} $$
We multiply by the power and subtract $1$ from the power.
For example, $$ \begin{equation}\begin{aligned} y&=x^2\\ \end{aligned}\end{equation} $$
produces the derivative $$ \begin{equation}\begin{aligned} \frac{dy}{dx}&=2\times x^{2-1}\\ &=2x^1\\ &=2x\\ \end{aligned}\end{equation} $$
Consider the example of a term with a coefficient: $$ \begin{equation}\begin{aligned} y&=4x^7\\ \end{aligned}\end{equation} $$ The derivative will be $$ \begin{equation}\begin{aligned} \frac{dy}{dx}&=7\times 4x^{7-1}\\ &=28x^6\\ \end{aligned}\end{equation} $$
For constants where the power of $x$ is $0$, the term disappears. For example, $$ \begin{equation}\begin{aligned} y&=7\\ &=7x^0\\ \end{aligned}\end{equation} $$ gives us $$ \begin{equation}\begin{aligned} \frac{dy}{dx}&=0\times 7x^{0-1}\\ &=0\\ \end{aligned}\end{equation} $$
For the sum/difference of many algebraic terms, the derivative is simply the sum/difference of the individual derivatives. Simply put, we differentiate each term. Given the function: $$ \begin{equation}\begin{aligned} y=x^5+3x^3-8\\ \end{aligned}\end{equation} $$
The derivative will be: $$ \begin{equation}\begin{aligned} \frac{dy}{dx}&=\frac{d}{dx}(x^5)+\frac{d}{dx}(3x^3)-\frac{d}{dx}(8)\\ &=5\times x^{5-1} + 3\times 3x^{3-1} - 0\times 8x^{0-1}\\ &=5x^4 + 9x^2\\ \end{aligned}\end{equation} $$
Practice
Find the derivatives of the following:
- $y=x^4$
- $y=3x^7$
- $y=\frac{1}{x^3}$
- $y=\sqrt{x}$
- $y=x^9-11x^2+3x$
- $y=2x^3-5\sqrt{x}+1$
- $y=\frac{-1}{x^5}+\frac{5}{x^2}$
Differentiating trigonometric functions
The following hold true: $$ \begin{equation}\begin{aligned} \frac{d}{dx}(\sin{x}) &= \cos{x}\\ \frac{d}{dx}(\cos{x}) &= -\sin{x}\\ \frac{d}{dx}(\tan{x}) &= \sec^2{x}\\ \end{aligned}\end{equation} $$
Chain rule
This rule is used when we have the composition of two functions.