One of the important operations you do in calculus is finding derivatives. The derivative of a function is also called the rate of change of that function. For instance, if x(t) is the position of a car at any time t, then the derivative of x, which is written dx/dt, is the velocity of the car. Also, the derivative can be visualized as the slope of a line tangent to the graph of a function. At a theoretical level, this is how mathematicians find derivatives. In practice, mathematicians use sets of basic rules and lookup tables.
The Derivative as a Slope
The slope of a line between two points is the rise, or difference in y values divided by the run, or difference in x values. The slope of a function y(x) for a certain value of x is defined to be the slope of a line that is tangent to the function at the point [x, y(x)]. To calculate the slope you construct a line between the point [x, y(x)] and a nearby point [x+h, y(x+h)], where h is a very small number. For this line, the run, or change in x value is h, and the rise, or change in y value, is y(x+h) - y(x). Consequently, slope of y(x) at the point [x,y(x)] is approximately equal to [y(x+h) - y(x)]/[(x + h) - x] = [y(x + h) - y(x)]/h. To get the slope exactly, you calculate the value of the slope as h gets smaller and smaller, to the “limit” where it goes to zero. The slope calculated this way is the derivative of y(x), which is written as y’(x) or dy/dx.
The Derivative of a Power Function
You can use the slope/limit method to calculate the derivatives of functions where y equals x to the power of a, or y(x) = x^a. For instance, if y equals x cubed, y(x) = x^3, then dy/dx is the limit as h goes to zero of [(x + h)^3 - x^3]/h. Expanding (x+h)^3 gives [x^3 + 3x^2h + 3xh^2 + h^3 - x^3]/h, which reduces to 3x^2 + 3xh^2 + h^2 after you divide by h. In the limit as h goes to zero, all terms that have h in them also go to zero. So, y’(x) = dy/dx = 3x^2. You can do this for values of a other than 3, and in general, you can show that d/dx(x^a) = (a - 1)x^(a-1).
Derivative From a Power Series
Many functions can be written as what are called a power series, which are the sum of an infinite number terms, where each is of the form C(n)x^n, where x is a variable, n is an integer and C(n) is a specific number for each value of n. For instance, the power series for the sine function is Sin(x) = x - x^3/6 + x^5/120 - x^7/5040 + ..., where “...” means terms continuing on to infinity. If you know the power series for a function, you can use the derivative of the power x^n to calculate the function’s derivative. For example, the derivative of Sin(x) is equal to 1 - x^2/2 + x^4/24 - x^6/720 + ..., which happens to be the power series for Cos(x).
Derivatives From Tables
The derivatives of basic functions such as powers like x^a, exponential functions, log functions and trig functions, are found using the slope/limit method, the power series method or other methods. These derivatives are then listed in tables. For instance, you can look up that the derivative of Sin(x) is Cos(x). When complex functions are combinations of the basic functions, you need special rules such as the chain rule and product rule, which are also given in the tables. For instance, you use the chain rule to find that the derivative of Sin(x^2) is 2xCos(x^2). You use the product rule to find that the derivative of xSin(x) is xCos(x) + Sin(x). Using tables and simple rules, you can find the derivative of any function. But when a function is extremely complex, scientists sometimes resort to computer programs for help.