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. If the function y is a natural log of a function of y, then you use the log rule and the chain rule.
For example, If the function is:. Then we apply the chain rule , first by identifying the parts:. Note that the generalized natural log rule is a special case of the chain rule :.
Taking the derivative of an exponential function is also a special case of the chain rule. First, let's start with a simple exponent and its derivative. When a function takes the logarithmic form:. No, it's not a misprint! The derivative of e x is e x. Just as a first derivative gives the slope or rate of change of a function, a higher order derivative gives the rate of change of the previous derivative.
We'll tak more about how this fits into economic analysis in a future section, [link: economic interpretation of higher order derivatives] but for now, we'll just define the technique and then describe the behavior with a few simple examples. To find a higher order derivative, simply reapply the rules of differentiation to the previous derivative. For example, suppose you have the following function:. Okay this may sound stupid but I need a little help You can apply this operator to a differentiable function.
And you get a new function. Do you have any concrete examples for which you need to calculate these two? It would probably make it more easy to grasp for you if I could explain it in a few examples. Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Ask Question. For example, what if any false statements and wrong formulas will it lead to? Of course you don't want to tell the students that, but it does clear up the logical question as asked. I am fine with using the notion of cancellation of fractions to help students remember the chain rule, but it is dangerous to be too cavalier with this idea.
Then I ask if anyone can say what step in the argument might be questionable. And then I point out that this is another use of the chain rule. I happen to talk here on this point of view. I always explain in terms of linear approximation. The infinitesimal point of view is useful in math an physics. I also recommend Infinitesimal Calculus by James M. Henle and Eugene M. You can introduce e. No mucking around with the limit definition to get these results, just elementary analytic geometry.
What's most misleading about Leibnizian notation is its implicit context dependence. I think it is important for calculus students to get the idea that differentiation is an operation that takes one function and produces a new function.
In that way, it is fundamentally different from addition or unary negation of numbers which is not the same thing as addition of functions. Note that I am a lot more interested in theoretical computer science than any form of physics - this may bias my point of view.
I just wish to share, that when I was an undergrad student I felt pretty much satisfied reading the introductory chapters of the ODE book by Arnol'd. For me, personally, it was pretty enjoyable, as I was also a bit dazed by the notion of fractions of infinitesimally small quantities. A note from a publication my own that occurred several years after this question was asked. You can think of differentials as infinitesimal values that are related to each other.
You can see an example of it in action here:. This separate use of dx and dy is particularly common in integration, which is the inverse of differentiation, but is also used in other ways. But certain things you can do with a derivative look an awful lot like what we can do with fractions:.
And the rules for substitution in an integral are proved by the chain rule, not by treating differentials as real entities. One place where we actually write differentials on their own, besides integration, is in estimation , which ties in with the theory of differentials:. In our intuitive idea of the meaning of the derivative, we can think of dx and dy as infinitesimal very tiny changes in the variables; here, we allow them to have any size, while recognizing that our estimate will not be very good if they get too large.
The second derivative notation comes from once again imagining that the derivative is really a fraction:. As we saw earlier, we can put the variable or function either within the fractional notation, or outside of it. Your email address will not be published. This site uses Akismet to reduce spam. Learn how your comment data is processed.
Skip to content Last week we looked at the meaning of the derivative. Hello, I am currently studying calculus, and am doing derivatives. I know how to take derivatives of various functions, but I am confused about the notation. Can you please give me a general overview of the meanings of the different derivative notations? But we think of those changes as being very, very small -- just looking at the limit.
0コメント