A derivative of a function is the slope or rate of change of a function. The proper notation for that is dy/dx(f(x)). This would be read as ‘dee-y dee-x of f of x’; this is just to shorten the notation instead of the ‘derivative of y derivative of x of the function f of x’. This notation of ‘d’ is just stating that this particular variable is the rate of change and not a particular variable.
In most occasions when working with derivatives one should only need to know four rules. The constant rule, the power rule, the product rule, and the quotient rule.
The constant rule states that the derivative of a constant number is 0 Ex: dy/dx(7) = 0; this is because the function of 7 would be y=7 which is a horizontal line 7 units above the x-axis which makes the slope is 0. This would work with any constant number because those functions simply do not change with x.
The power rule states that dy/dx(x^n)=n*x^(n-1). This describes how whenever we have a variable to a power we multiply that power by our leading coefficient and subtract that power by 1. An example of this would be dy/dx(x^3)=3*x^2.
The product rule states that dy/dx(uv) = u(dv)+v(du)=uv′+vu′; where u and v are different functions or variables. With this example one can make several observations one of which is that the du and dv can be replaced with u′ and v′. These are pronounced as ‘u prime’ and ‘v prime’ and represent the rate of change of these variables. One should also recongnize that the multiplication of these functions leads to addition. This is a straight forward rule but it is very similar to thinking that the multiplication of powers leads to their addition Ex: x^2*x^3=x^(2+3)=x^5 and is a good way to remember how this rule works.
The quotient rule states that dy/dx(u/v)= (u′v-v′u)/(v^2); where u and v are different functions or variables. This example is very similar to the last but it is just the opposite because the ways that powers change and dividing would result in a negative.
In all cases there are only 14 rules of derivatives but some are very uncommon and would not be worked with that often. In almost any case one would be working with these 4 rules because these are the fundamental principles that apply to many functions and will be used almost every single time they are taking the derivative of a function.
Some may ask ‘do I need the equation of a function to find its derivative?’ This answer is a little complex but basically the derivative of a function is defined as the rate of change. When working with equations most of the time the slope or rate of change will change due to the curving nature of the function; because if for every x the slope was the same it would be a linear line. So without the equation of a function we cannot find the slope at every single point but we can calculate the slope of the function at one point of a function.
In other words if we had a point (5,3) that was between (4,1) and (6,5) we could calculate the slope by rise-over run. (5-1)/(6-4)=4/2=2. This means that when x is between 4 and 6 (4<x<6) the slope is 4 and that means the derivative of the function is 4. One could also imagine that this representation is a ‘Tangent Line’ or line that is parallel to the function of the graph at the point 5. This may not seem very foolproof but it can be found useful when you need a derivative at a specific point to plug into an implicit differential or related rates problem. This method could also be used when a problem solver is just given the graph of a function and they have to find the points themselves; when it is not needed to actually take the derivative this is a very quick and easy method. Finding a derivative may seem complicated at first but with these four basic rules and an understanding that derivative=slope=rate of change one should be able to accurately find the derivative of a vast amount of general equations.