Active 1 month ago. Viewed 88k times. I need a thorough explanation. Kami Kami 1 1 gold badge 2 2 silver badges 5 5 bronze badges. These symbols are derivatives.
Are you familiar with derivatives? Add a comment. Active Oldest Votes. Thomas Thomas It is not a fraction. It's the limit of a fraction. It is nevertheless CAN be defined as a fraction of two functions, rather than an atomic object. I hope you understand. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. As a pure trading platform, dYdX is quite limited, but as a completely open, trustless, and non-custodial financial protocol, it is one of the most advanced.
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If the total function is f minus g, then the derivative is the derivative of the f term minus the derivative of the g term. The most straightforward approach would be to multiply out the two terms, then take the derivative of the resulting polynomial according to the above rules. Or you have the option of applying the following rule.
Read this as follows: the derivative of y with respect to x is the derivative of the f term multiplied by the g term, plus the derivative of the g term multiplied by the f term. The quotient rule is similarly applied to functions where the f and g terms are a quotient.
Then follow this rule:. Now, let's combine rules by type of function and their corresponding graphs. There are two more rules that you are likely to encounter in your economics studies. The hardest part of these rules is identifying to which parts of the functions the rules apply. Actually applying the rule is a simple matter of substituting in and multiplying through.
Notice that the two rules of this section build upon the rules from the previous section, and provide you with ways to deal with increasingly complicated functions, while still using the same techniques. In the previous rules, we dealt with powers attached to a single variable, such as x 2 , or x 5.
Suppose, however, that your equation carries more than just the single variable x to a power. For example,. Then the problem becomes. Now, note that your goal is still to take the derivative of y with respect to x. However, x is being operated on by two functions; first by g multiplies x by 2 and adds to 3 , and then that result is carried to the power of four.
Therefore, when we take the derivatives, we have to account for both operations on x. First, use the power rule from the table above to get:. Note that the rule was applied to g x as a whole. Note the change in notation. Now, both parts are multiplied to get the final result:. Recall that derivatives are defined as being a function of x. Then simplify by combining the coefficients 4 and 2, and changing the power to The second rule in this section is actually just a generalization of the above power rule.
It is used when x is operated on more than once, but it isn't limited only to cases involving powers. Since you already understand the above problem, let's redo it using the chain rule, so you can focus on the technique. This type of function is also known as a composite function.
The derivative of a composite function is equal to the derivative of y with respect to u, times the derivative of u with respect to x:. Recall that a derivative is defined as a function of x, not u. The formal chain rule is as follows. When a function takes the following form:. There are two special cases of derivative rules that apply to functions that are used frequently in economic analysis.
You may want to review the sections on natural logarithmic functions and graphs and exponential functions and graphs before starting this section.
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