1=2 d dx x 1 x+ 2! If , where u is a differentiable function of x and n is a rational number, then Examples: Find the derivative of each function given below. 1. y=f(u) u=f(x) y=(2x+4)3 y=u3andu=2x+4 dy du =3u2 du dx =2 dy dx It is useful when finding the derivative of a function that is raised to the nth power. Let’s walk through the solution of this exercise slowly so we don’t make any mistakes. EXAMPLE 2: CHAIN RULE Step 1: Identify the outer and inner functions Example: Chain rule for f(x,y) when y is a function of x The heading says it all: we want to know how f(x,y)changeswhenx and y change but there is really only one independent variable, say x,andy is a function of x. Lecture 3: Chain Rules and Inequalities Last lecture: entropy and mutual information This time { Chain rules { Jensen’s inequality { Log-sum inequality { Concavity of entropy { Convex/concavity of mutual information Dr. Yao Xie, ECE587, Information Theory, Duke University VCE Maths Methods - Chain, Product & Quotient Rules The chain rule 3 • The chain rule is used to di!erentiate a function that has a function within it. y c CA9l5l W ur Yimgh1tTs y mr6e Os5eVr3vkejdW.I d 2Mvatdte I Nw5intkhZ oI5n 1fFivnNiVtvev 4C 3atlyc Ru2l Wu7s1.2 Worksheet by Kuta Software LLC Example 5.6.0.4 2. By the chain rule, F0(x) = 1 2 (x2 + x+ 1) 3=2(2x+ 1) = (2x+ 1) 2(x2 + x+ 1)3=2: Example Find the derivative of L(x) = q x 1 x+2. Let Then 2. √ √Let √ inside outside In such a case, we can find the derivative of with respect to by direct substitution, so that is written as a function of only, or we may use a form of the Chain Rule for multi-variable functions to find this derivative. I Chain rule for change of coordinates in a plane. Example: Differentiate y = (2x + 1) 5 (x 3 – x +1) 4. EXAMPLE 2: CHAIN RULE A biologist must use the chain rule to determine how fast a given bacteria population is growing at a given point in time t days later. Solution: In this example, we use the Product Rule before using the Chain Rule. Here we use the chain rule followed by the quotient rule. • The chain rule • Questions 2. This 105. is captured by the third of the four branch diagrams on … The population grows at a rate of : y(t) =1000e5t-300. I Functions of two variables, f : D ⊂ R2 → R. I Chain rule for functions defined on a curve in a plane. Example 4: Find the derivative of f(x) = ln(sin(x2)). Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. ©T M2G0j1f3 F XKTuvt3a n iS po Qf2t9wOaRrte m HLNL4CF. Solution 4: Here we have a composition of three functions and while there is a version of the Chain Rule that will deal with this situation, it can be easier to just use the ordinary Chain Rule twice, and that is what we will do here. (x) The chain rule says that when we take the derivative of one function composed with The chain rule is the most important and powerful theorem about derivatives. 1=2: Using the chain rule, we get L0(x) = 1 2 x 1 x+ 2! For a first look at it, let’s approach the last example of last week’s lecture in a different way: Exercise 3.3.11 (revisited and shortened) A stone is dropped into a lake, creating a cir-cular ripple that travels outward at a … 14.4) I Review: Chain rule for f : D ⊂ R → R. I Chain rule for change of coordinates in a line. Chain rule for functions of 2, 3 variables (Sect. In applying the Chain Rule, think of the opposite function f °g as having an inside and an outside part: General Power Rule a special case of the Chain Rule. 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