By the product rule we obtain. Rearranging this statement and dividing by h yields. Exercise 8. Apply it to find. To find the derivative of the inverse function to h x , you need only to observe that the inverse function is obtained by switching x and y axes; since the derivative of h is the slope of the tangent line of its graph, after switching the h and x axes we get slope.
Thus the derivative of the inverse function h -1 x to h at argument h x is the reciprocal of the derivative of h with respect to x and argument x.
This sounds worse than it is. This is actually the general idea used above to evaluate the derivatives both of and h -1 the reciprocal and the inverse functions to h. Notice that there are really only two rules that we have invoked here which allow us to differentiate all standard functions. One is the multiple occurrence rule, which allows us to treat separate occurrences of a variable separately and add their individual derivatives up to get the whole derivative.
The second is the chain rule, which notes that the derivative is a slope which is a ratio of changes, so that changing the independent variable, which changes the denominator of the slope, requires a change in the derivative of the ratio of the old denominator to the new one. Add to the derivative of the constant which is 0, and the total derivative is 15x 2.
Note that we don't yet know the slope, but rather the formula for the slope. These rules cover all polynomials, and now we add a few rules to deal with other types of nonlinear functions. It is not as obvious why the application of the rest of the rules still results in finding a function for the slope, and in a regular calculus class you would prove this to yourself repeatedly. Here, we want to focus on the economic application of calculus, so we'll take Newton's word for it that the rules work, memorize a few, and get on with the economics!
The most important step for the remainder of the rules is to properly identify the form, or how the terms are combined, and then the application of the rule is straightforward. For functions that are sums or differences of terms, we can formalize the strategy above as follows:. Here's a chance to practice reading the symbols. Read this rule as: if y is equal to the sum of two terms or functions, both of which depend upon x, then the function of the slope is equal to the sum of the derivatives of the two terms.
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.
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.
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