What is dy over dx

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:. According to our rules, we can find the formula for the slope by taking the first derivative:. If we need a third derivative, we differentiate the second derivative, and so on for each successive derivative. Note that the notation for second derivative is created by adding a second prime.

Other notations are also based on the corresponding first derivative form. Here are some examples of the most common notations for derivatives and higher order derivatives. Now for some examples of what a higher order derivative actually is. Let's start with a nonlinear function and take a first and second derivative. Recall from previous sections that this equation will graph as a parabola that opens downward [link: graphing binomial functions].

In order to understand the meaning of derivatives, let's pick a couple of values of x, and calculate the value of the derivatives at those points. So, how do we interpret this information? When x equals 0, we know that the slope of the function, or rate of change in y for a given change in x from the first derivative is 6.

Similarly, the second derivative tells us that the rate of change of the first derivative for a given change in x is In other words, when x changes, we expect the slope to change by -2, or to decrease by 2. We can check this by changing x from 0 to 1, and noting that the slope did change from 6 to 4, therefore decreasing by 2.

To sum up, the first derivative gives us the slope, and the second derivative gives the change in the slope. In economics, the first two derivatives will be the most useful, so we'll stop there for now. Derivative is a function, actual slope depends upon location ie value of x. Click here for the answer. Using Dot Style If you graph the parametric equations using Dot style the diagonal lines will not be displayed.

Module 25 - Parametric Equations. Lesson Finding Derivatives from a Graph The derivatives at a point on the graph of a parametric curve can be found by using the derivative features of the CALC menu on the Graph screen.

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. For equations, use MathJax. Sign up or log in Sign up using Google.

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