Product Rule and Power Rule Derivatives


    Today at online calculus we are going to be checking out two of the most important differentiation rules, the product rule and the power rule. Up until now we have only seen very basic differentiation rules, like the power and exponential rules. It is now time to learn a more complicated rule for differentiating the product or two functions. Keep in mind that you need good graphing calculator for this. You can read the review of the new graphing calculator TI-84 Plus CE from Texas Instruments.

    Product Rule Derivatives

    There are simple rules for taking the derivative of a sum or difference of two functions (the sum and difference rules), but it is slightly more complicated when it comes to multiplication and division. If we want to find the derivative of a functions that consists of two other function multiplied together, we cannot simply take the derivative of each function in the product and multiply them together afterwards.

    This will give a different (and wrong) result. In simple language, the product rule is; the derivative of the first times the second, plus the derivative of the second times the first. So if we have two functions, f(x) and g(x) then;[f(x)g(x)]’= f’(x)g(x) + g’(x)f(x). It is an important rule to memorize as you will be using it a lot in the future.

    For example, someone asks us to find the derivative of the function y = x^2 * e^x. We need to use the product rule to solve this. We will break the problem down into steps to make it easy for ourselves. First, we find the derivative of the first term (x^2) to be 2x using the power rule. We also know the derivative of the second (e^x) is simply e^x (this is that important rule from the previous section).

    So now we take these terms and put them into the product rule.y’ = 2x * e^x + e^x * x^2We can factor this answer but it is not necessary. This is a perfectly acceptable final answer. The product rule is not difficult to use if you break it into specific steps. Find the derivative of the first, the second, and the put them into the product rule.

    Power Rule Derivatives

    Using differentiation rules will greatly simplify the process of  finding a derivative. There are many rules to remember so be sure to practice  them often. The first rule to remember is that the derivative of any  constant is simply 0.

    So if you take the derivative of any number without a  variable, the answer will be 0. Why is that? Well recall that the derivative is  the rate of change of the function. A constant does not change. It is simply a  straight line on a graph, so therefore it will have a zero value derivative for  all of x.

    The power rule is one of the most commonly used rules in  differential calculus. The power rule can be written as; (d/dx)x^n=nx^(n-1). It  is a fairly simple procedure, but be sure to practise it. You simply take the  power of x, and bring it in front of x. Then you subtract one from the power of  x.

    For example the derivate of x^2 is simply 2x. The derivative of x^3 is 3x^2.  See the video for a better explanation of the procedure. Again, for example, someone asks us to find the  derivative of f(x)=sqrt(x^3). Before doing anything we want to simplify this  equation and write the function as f(x)=x^1.5. So now by using the power rule,  we bring the 1.5 out front. Then we subtract 1 from the power. So we are left  with f(x)=1.5x^0.5 which is the same as 1.5*sqrt(x).

    If you are having difficulties with these rules, just remember, practice makes perfect!