Well natural log of a, it might not immediately jump out to you, but that's just going to be a number. So that's just going to be, so times the derivative. If it was the derivative of three x, it would just be three.
If it's the derivative of natural log a times x, it's just going to be natural log of a. And so this is going to give us the natural log of a times e to the natural log of a.
And I'm going to write it like this. Natural log of a to the x power. Well we've already seen this. This right over here is just a. So it all simplifies. It all simplifies to the natural log of a times a to the x, which is a pretty neat result. So if you're taking the derivative of e to the x, it's just going to be e to the x. If you're taking the derivative of a to the x, it's just going to be the natural log of a times a to the x.
And so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. So if I want to find the derivative with respect to x of eight times three to the x power, well what's that going to be?
Well that's just going to be eight times and then the derivative of this right over here is going to be, based on what we just saw, it's going to be the natural log of our base, natural log of three times three to the x.
Show 6 more comments. Cow S. Cow 9 9 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. How do we interpret this? For example, suppose you would like to know the slope of y when the variable x takes on a value of 2. Now for the practical part. How do we actually determine the function of the slope? Almost all functions you will see in economics can be differentiated using a fairly short list of rules or formulas, which will be presented in the next several sections.
Once you understand that differentiation is the process of finding the function of the slope, the actual application of the rules is straightforward. First, some overall strategy. The rules are applied to each term within a function separately. Then the results from the differentiation of each term are added together, being careful to preserve signs. Don't forget that a term such as "x" has a coefficient of positive one. Coefficients and signs must be correctly carried through all operations, especially in differentiation.
The rules of differentiation are cumulative, in the sense that the more parts a function has, the more rules that have to be applied. Let's start here with some specific examples, and then the general rules will be presented in table form. The derivative of any constant term is 0, according to our first rule. This makes sense since slope is defined as the change in the y variable for a given change in the x variable.
Suppose x goes from 10 to 11; y is still equal to 15 in this function, and does not change, therefore the slope is 0. Note that this function graphs as a horizontal line. The next rule states that when the x is to the power of one, the slope is the coefficient on that x. This continues to make sense, since a change in x is multiplied by 2 to determine the resulting change in y. We add this to the derivative of the constant, which is 0 by our previous rule, and the slope of the total function is 2.
Now, suppose that the variable is carried to some higher power. The power rule combined with the coefficient rule is used as follows: pull out the coefficient, multiply it by the power of x, then multiply that term by x, carried to the power of n - 1. Therefore, the derivative of 5x 3 is equal to 5 3 x 3 - 1 ; simplify to get 15x 2. 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. Collectively, these are referred to as higher-order derivatives. Learning Objectives Define the derivative function of a given function. Graph a derivative function from the graph of a given function. State the connection between derivatives and continuity. Describe three conditions for when a function does not have a derivative.
Explain the meaning of a higher-order derivative. Derivative Functions The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. Solution Start directly with the definition of the derivative function. Solution Follow the same procedure here, but without having to multiply by the conjugate.
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