![]() To know how much your profits are increasing at time t, you need to multiply that rate by the profit per sale. That is, at time t, your sales are increasing one unit per second. This means that sales are increasing and profits per sale also are increasing.Ī concrete value of u'(t) could be one tomato per second. Let's say, for the sake of simplicity, that both u'(t) and v'(t) are positive for a specific time t. This means that S'(t) will be a function of the rates of change of those functions: u'(t) and v'(t). For your profit to change, at least one of two things must change: your sales or your profits per sale. S'(t) is the rate of change of your profit. In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. So, let's try to solve the problem of finding S'(t). The rate of change of your profit is valuable information for you. ![]() And the tool for that is the derivative.Īs we learned in the page on the intuitive idea behind the derivative, the derivative gives the rate of change of a function. A natural question you make as a business owner is: what is the trend? You want to predict how your profits will change in the future. ![]() Now, we have the profits S(t) expressed as a profuct. I think you do understand Sals (AKA the most common) proof of the product rule. It is obvious why u(t) changes: you sell more or less tomatoes depending of the time t.īut v(t) also changes: the market price of tomatoes and your production costs change all the time, hence your profits per sale change with time. It is this type of insight and intuition, that being, the ability to leverage the rules of mathematics creatively that produces much of the beauty in math. Where u(t) is how many tomatoes you sold at time t, and v(t) is your profit for each tomato sold at time t. In this case we can express your profit S(t) as a (mathematical) product. In this case, the graph of S(t) against time would be a continuous curve.Īlso, let's say that your business sells a single product: tomatoes. Then, they make a sale and S(t) makes an instant jump.įor the sake of this explanation, let's say that you business is huge, making sales every second of the day. For many businesses, S(t) will be zero most of the time: they don't make a sale for a while. There is nothing stopping us from considering S(t) at any time t, though. For example, your profit in the year 2015, or your profits last month. We usually think of profits in discrete time frames. A function S(t) represents your profits at a specified time t. Let's say you are running a business, and you are tracking your profits. ![]() Here we'll first focus on trying to get an intuitive understanding of why this rule is the way it is using simple examples. Then we'll solve some example problems applying it. You can easily find that on other websites. The product rule is one of the essential differentiation rules. In any calculus textbook the introduction to this rule is a formal deduction using the definition of the derivative. ![]()
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