Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
Working with Derivatives
Suppose you have a job for a company that builds cars. Your boss tells you they are working on a car part, and a certain aspect of it can be represented by one of the following four functions.
You're told that you need to be able to find the slope of each of these functions for any given value of x. You remember that the derivative of a function represents the slope of that function at any given point. Great! All you have to do is find the derivative of each of these functions and your boss will be a happy camper.
You look at each of the functions and realize that the first function is a sum of functions, the second function is a difference of functions, the third function is a product of functions, and the fourth function is a quotient of functions. Thus, we want to look at how to find the derivative of a sum, difference, product and quotient. Let's get started!
Derivative of a Sum
When calculating the derivative of a sum, we simply take the sum of the derivatives. This is illustrated in the following formula.
The first function that your boss wants you to work with is the sum of two functions. Therefore, to find the derivative of this function, we just take the sum of the derivatives. To do this, we need to recognize that the derivative of x 2 is 2x, and the derivative of 4x is 4. Now we just plug into our formula and we've got our derivative.
We see that the derivative of the first function is f ' (x) = 2x + 4.
Derivative of a Difference
The second function, g, is a difference of functions. When we want to find the derivative of a difference, we simply find the difference of the derivatives. This is similar to the sum rule, and is shown in the following formula.
To use our formula for our example, we just need to know that the derivative of 3x3 is 9x 2, and the derivative of 5x 2 is 10x. Once again, we just plug into our formula.
Using our formula, we've found that the derivative of the second function is g' (x) = 9x 2 - 10x.
Derivative of a Product
Now things get a little trickier. The third function is a product of the functions x and ln(x). When we want to find the derivative of a product, we use the product rule for derivatives.
Thus, to find the derivative of our function h in our example, we need to know that the derivative of x is 1, and the derivative of ln(x) is 1/x, and then we just need to use our formula.
We see that the derivative of the third function is h' (x) = ln(x) + 1.
Derivative of a Quotient
The last function is a quotient. Once again, taking the derivative of a quotient is a little trickier than just taking the quotient of the derivatives. We have a quotient rule for derivatives as well, and it is as follows.
To use the quotient rule to find the derivative of j, we need to know the derivative of x and ln(x). We actually already know these from the last problem. The derivative of x is 1, and the derivative of ln(x) is 1/x. All we have to do now is plug into our formula and simplify.
We've found the derivative of the last function to be j ' (x) = (1 - ln(x)) / x 2.
We've now found all the derivatives of the functions that your boss gave you, so now he can find the slope of each of those functions at any given value of x. Great job! You should ask for a raise!
When it comes to finding the derivative of a sum, difference, product, or quotient, we have different rules for each case. These rules are as follows.
When we want to use one of these formulas, we simply calculate the different parts of the formula and then plug in and simplify. We can see that these formulas make finding these derivatives fairly easy, so it is a good idea to try to put these to memory for future use.
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