Functions are rules that clarify the relationship of an independent variable and a dependent variable. If a number is assigned to an independent variable and a value is then found for the dependent variable because of a rule, then the dependent variable is a function of the independent variable. The common equation for functions is represented as y = f(x), where y, the dependent variable, is equal to the function of x, the independent variable. Many formulas are written as functions, including the formula for finding area. There are also polynomial functions where a function of x, or f(x), will equal a long mathematical equation in which the variable x appears multiple times.
Derivatives of functions are written as f'(x). A derivative is actually the limit of the difference quotient of a function. The difference quotient is the measure of a secant line that passes through two definitive points, often called a and b, on a graphically represented function. Very simple functions such as f(x) = x will have a derivative of 1 because a single variable will yield a single horizontal line on a graph. A secant line cannot be drawn between two points of a single line, rendering the difference quotient nonexistent and making the derivative 1. Aside from finding the limit of the difference quotient, which is a very lengthy equation, you can also use several rules of calculus to find the derivative of a function.
The power rule is used with single variable functions --- some number x to the power of some number n, or x^n. The power rule simply demonstrates that you can move that number n in front of the x and subtract the superscript n by one. This will look something like the equation nx^n - 1, where n - 1 is the superscript. If the x does not have a superscript, you should imagine that the superscript is actually the number 1, because x to the power of 1 is still x. If you imagine that it is the number 1, you get 1x^1 - 1, or 1 multiplied by x to the zero power. The resulting derivative of f(x) = x is then 1, which matches the definition of derivatives from above. This is a simple way of finding derivatives, because it only involves rearranging numbers around the variable.
The chain rule finds the derivative of multiple variables. For instance, if you have that a function, F(x), is equal to f(g(x)), you have a function equal to a function within a function. This sounds complicated but, by using the chain rule, you can easily rearrange this equation to find the derivative. The key is to take this as two functions. The derivative of the first function is f'(g(x)), and the second derivative is g'(x). The chain rule states to split the first function into these two functions and then multiply their derivatives together. After you have split the functions, you can use the power rule to find the derivatives.