When to Use the Chain and Quotient RulesDate: 06/12/98 at 23:35:42 From: Matthew Subject: Order of differentiation Hi, I'm doing a maths subject called "Introductory Calculus" and learning about the various forms of differentiation. I've come across such questions as: Differentiate: (2x + 1)^2 f(x) = ---------- 2x + 4 In situations like this, do I use the quotient rule first or the chain rule first? Date: 06/16/98 at 23:06:37 From: Doctor Mateo Subject: Re: Order of differentiation Hello Matthew, In the specific example that you give, you begin by recognizing that the function is a rational function - it looks like a fraction with a variable in the denominator (the bottom). Because you are looking for the first derivative of a rational function, you begin by applying the quotient rule first. As you apply the quotient rule, you encounter a term in the numerator with a power. So what happens here is that you end up applying the chain rule in the process of doing the quotient rule. Consider the two functions below: (4x - 5)^3 ( (4x - 5) )^3 (I) g(x) = ----------- and (II) h(x) = (----------) (5 + 3x) ( (5 + 3x) ) In I) above, the exponent is part of the term in the numerator (top). The exponent does not go with the term in the denominator. In II) above, the exponent is on the outside of the entire fraction, which means that it actually belongs to the numerator (top) and the denominator (bottom) because: ( (4x - 5) )^3 (4x - 5)^3 h(x) = (----------) = ---------- ( (5 + 3x) ) (5 + 3x)^3 In II) you would apply the chain rule first, since the exponent goes with the numerator and the denominator. Then you would do the quotient rule on the rational function inside. Alternatively, you could put the exponent with the numerator and the denominator then apply the quotient rule first, since you now have a rational function, but then you would have to use the chain rule twice. In I) you would apply the quotient rule first since you have a rational function, and then do the chain rule on the part that involves the exponent. I hope this helps you distinguish the order a little better. Have fun differentiating! -Doctor Mateo, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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