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For example, y = cosx x2 We write this as y = u v where we identify u as cosx and v as x2. There is a formula we can use to differentiate a quotient - it is called thequotientrule. In this unit we will state and use the quotient rule. 2. The quotient rule The rule states: Key Point Thequotientrule:if y = u v then dy dx = vdu dx −udv v2
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Example 20.1 Find the derivative of 4x3ex. This is a product (4x3)·(ex of two functions, so we use the product rule. Dx h 4x3ex i = Dx £ 4x3 § ·ex +4x3 ·Dx £ ex § = 12x2 ·ex +4x3 ·ex = 4ex ° 3x2 +x3 ¢. Example 20.2 Find the derivative of y= ° x2 +3 ¢° 5 °7 ¢. This is a product of two functions, so we use the product rule. Dx h ...
If we need to take the derivative of two functions being divided, we cannot simply divide the derivative of the numerator by the derivative of the denominator; d dx f(x) g(x) 6= f0(x) g0(x): Example 1: Compute the derivative of the following function. y = sin(x)+x 2x+1 Example 2: Compute the derivative of the following function. y = aex (a2 + p x)
Quotient rule: f g 0 = f 0g fg g2 For our example to illustrate the use of the product rule, let’s nd the derivative of x p x. Here, f(x) = x while g(x) = p x. Then (x p x)0 = f0(x)g(x) + f(x)g0(x) = 1 p x+ x 1 2 p x and this last expression simpli es to 3 2 p x. What’s important to see in this example is how to use the product rule.
First, f(cx) = m(cx) = c(mx) = cf(x), so the constant c can be “moved outside” or “moved through” the function f. Second, f(x + y) = m(x + y) = mx + my = f(x) + f(y), so the addition symbol likewise can be moved through the function. The corresponding properties for the derivative are: (cf(x))′ =. cf(x) = c dx.
Use the quotient rule to find the minimum of AC then show that at this point AC = MC. Answer: x = 40 If we don’t get to this one during Lecture, do it yourself for practice using the quotient rule. Example J: Given ( ) 2 3 x x f x = , find f ′(x). In Lecture 1.3 Example G we simplified and then used the power rule to find ( ) 3 8 3 5 x f ...
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To differentiate such expressions we use the quotient rule, which can be written as: (a) We express in the form , so that . Giving the following derivatives, . Using the quotient rule we have, (b) First express in the form , so that and. Using the quotient rule, we have Function Derivative If then If then y x2 x3+x–1 = -----y u v
