The quotient rule is the last rule for differentiation that will be discussed in these worksheets. In what follows we explore why this is the case, what the product and quotient rules actually say, and work to expand our repertoire of functions we can easily differentiate. Final quiz solutions to exercises solutions to quizzes the full range of these packages and some instructions, should they be required, can be obtained from our web page mathematics support materials. It doesnt matter if you reverse the terms in the product rule, but it does matter in the. So hopefully this makes the product rule a little bit more tangible. For the love of physics walter lewin may 16, 2011 duration. Derivatives of exponential and logarithm functions. Ap calculus ab worksheet 22 derivatives power, package. As their names suggest, the product rule and the quotient rule are used to di. Theyre very useful because the product rule gives you the derivatives for the product of two functions, and the quotient rule does the same for the quotient of two functions. Selection file type icon file name description size revision time user. Learning objectives use the product rule to multiply exponential expressions with like bases. The quotient rule mctyquotient20091 a special rule, thequotientrule, exists for di.
The quotient rule states that for two functions, u and v, see if you can use the product rule and the chain rule on y uv 1 to derive this formula. The product rule and the quotient rule are a dynamic duo of differentiation problems. Power rule chain rule product and quotient rule dana ernst. This is the same thing as e to the x times cosine of x minus sine of x. Product rule, quotient rule jj ii product rule, quotient rule. The exponent rule for dividing exponential terms together is called the quotient rule. Simplify expressions using a combination of the properties. Quotient rule practice find the derivatives of the following rational functions. Before using the chain rule, lets multiply this out and then take the. In some cases it might be advantageous to simplifyrewrite first.
The proof of the quotient rule is shown in the proof of various derivative formulas section of the extras chapter. It might stretch your brain to keep track of where you are in this process. Functions to differentiate include polynomials, rationals, and radicals. In this section, you will learn how to find the derivative of a product of functions and the derivative of a quotient of functions. Use the quotient rule to divide exponential expressions with like bases. The quotient rule is actually the product rule in disguise and is used when differentiating a fraction. Use the quotient rule to find the derivative of \\displaystyle g\left x \right \frac6x22 x\. Furthermore, when we have products and quotients of. If the exponential terms have multiple bases, then you treat each base like a common term. First, we should discuss the concept of the composition of a function which actually means the function of another function. We must use the quotient rule, and in the middle of it, when we get to the part where we take the derivative of the top, we must use a product rule to calculate that. Find the first derivative of the following functions.
The product rule must be utilized when the derivative of the product of two functions is to be taken. Power rule, derivative the exponential function derivative of a sum di erentiability implies continuity. One might expect from this that the derivative of a product is the product of the derivatives. The proof of the product rule is shown in the proof of various derivative formulas. The product and quotient rules university of reading. Consider the product of two simple functions, say where and. The only prior knowledge required is the power rule. Differentiate using the product and quotient rules practice. The product rule mctyproduct20091 a special rule, theproductrule, exists for di. If our function f can be expressed as fx gx hx, where g and h are simpler functions, then the quotient rule may be stated as f. Now that we know where the power rule came from, lets practice using it to take derivatives of polynomials. If this confuses you, go back to the top of the page and reread the product rule and then go through some examples in your textbook.
The product and quotient rules university of plymouth. You can still go the long way on these problems and simplify by writing out all the factors and combining or. Or, if you want, you could factor out an e to the x. Use proper notation and simplify your final answers. Product rule we have seen that the derivative of a sum is the sum of the derivatives. Differentiate using the product and quotient rules. There isnt much to do here other than take the derivative using the quotient rule. Before you tackle some practice problems using these rules, heres a. Since every quotient can be written as a product, it is always possible to use the product rule to compute the derivative, though it is not always simpler. Mar 18, 2020 selection file type icon file name description size revision time user.
Listofderivativerules belowisalistofallthederivativeruleswewentoverinclass. First using the quotient rule and then using the product rule. Show solution there isnt much to do here other than take the derivative using the quotient rule. The rule for how to divide exponents expresses that while dividing exponential terms together with a similar base, you keep the base and subtract the exponents. Note that the numerator of the quotient rule is very similar to the product rule so be careful to not mix the two up. Watch the video lecture chain, product and quotient rule. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
Exponents and the product, quotient, and power rules. An obvious guess for the derivative of is the product of the derivatives. Train 8th grade students to rewrite each exponential expression as a single exponent with this set of pdf worksheets. The product and quotient rules mathematics libretexts. This lesson contains the following essential knowledge ek concepts for the ap calculus course. For the statement of these three rules, let f and g be two di erentiable functions. It is easier to discuss this concept in informal terms. Furthermore, when we have products and quotients of polynomials, we can take the.
We can check by rewriting and and doing the calculation in a way that is known to work. Quotient rule the quotient rule is used when we want to di. The quotient rule in words the quotient rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. The product rule and the quotient rule scool, the revision. The quotient rule for exponents states that when dividing exponential terms together with the same base, you keep the base the same and then subtract the exponents. Simplify by using the product, quotient, and power rules.
And thats all you need to know to use the product rule. Then apply the product rule in the first part of the numerator. Proofs of the product, reciprocal, and quotient rules math. Mar 07, 2018 now that we know where the power rule came from, lets practice using it to take derivatives of polynomials. In this topic, you will learn general rules that tell us how to differentiate products of functions, quotients of functions, and composite functions. Notice that this example has a product in the numerator of a quotient. Click here for an overview of all the eks in this course.