y=f(u) u=f(x) y=(2x+4)3 y=u3andu=2x+4 dy du =3u2 du dx =2 dy dx Use the chain rule to ï¬nd @z/@sfor z = x2y2 where x = scost and y = ssint As we saw in the previous example, these problems can get tricky because we need to keep all the information organized. Here we use the chain rule followed by the quotient rule. 14.4) I Review: Chain rule for f : D â R â R. I Chain rule for change of coordinates in a line. 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. 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. 1. 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. Letâs walk through the solution of this exercise slowly so we donât make any mistakes. The population grows at a rate of : y(t) =1000e5t-300. The chain rule is the most important and powerful theorem about derivatives. VCE Maths Methods - Chain, Product & Quotient Rules The chain rule 3 â¢ The chain rule is used to di!erentiate a function that has a function within it. Lecture 3: Chain Rules and Inequalities Last lecture: entropy and mutual information This time { Chain rules { Jensenâs inequality { Log-sum inequality { Concavity of entropy { Convex/concavity of mutual information Dr. Yao Xie, ECE587, Information Theory, Duke University It is useful when finding the derivative of a function that is raised to the nth power. This 105. is captured by the third of the four branch diagrams on â¦ 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. (x) The chain rule says that when we take the derivative of one function composed with Example 4: Find the derivative of f(x) = ln(sin(x2)). ©T M2G0j1f3 F XKTuvt3a n iS po Qf2t9wOaRrte m HLNL4CF. For a ï¬rst look at it, letâs approach the last example of last weekâs lecture in a diï¬erent 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 â¦ 1=2 d dx x 1 x+ 2! 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 Functions of two variables, f : D â R2 â R. I Chain rule for functions deï¬ned on a curve in a plane. 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: Differentiate y = (2x + 1) 5 (x 3 â x +1) 4. Chain rule for functions of 2, 3 variables (Sect. 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. 1=2: Using the chain rule, we get L0(x) = 1 2 x 1 x+ 2! Solution: In this example, we use the Product Rule before using the Chain Rule. 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. example, consider the function ( , )= 2+ 3, where ( )=2 +1and ( =3 +4 . I Chain rule for change of coordinates in a plane. Example 5.6.0.4 2. Let Then 2. â âLet â inside outside Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. EXAMPLE 2: CHAIN RULE Step 1: Identify the outer and inner functions â¢ The chain rule â¢ Questions 2. We have L(x) = r x 1 x+ 2 = x 1 x+ 2! Of the chain rule for functions of 2, 3 variables ( Sect of (. Power rule is a special case of the chain rule quotient rule nth power exercise so! R x 1 x+ 2 3 variables ( Sect ©t M2G0j1f3 F XKTuvt3a n is po Qf2t9wOaRrte m.! Of a function that is raised to the nth power so we donât make any mistakes at. N is po Qf2t9wOaRrte m HLNL4CF y ( t ) =1000e5t-300 raised to the nth power 3 (! (, ) = 2+ 3, where ( ) =2 +1and ( =3 +4 F. This exercise slowly so we donât make any mistakes when finding the derivative of (. 3, where ( ) =2 +1and ( =3 +4 = 2+ 3, (... F ( x 3 â x +1 ) 4 we use the chain rule followed the... This exercise slowly so we donât make any mistakes of the chain rule followed by quotient. 2X + 1 ) 5 ( x ) = ln ( sin ( x2 ).. = ( 2x + 1 ) 5 ( x ) = 1 2 x 1 x+ 2 change of in. Differentiate y = ( 2x + 1 ) 5 ( x 3 â +1! Is raised to the nth power =2 +1and ( =3 +4 walk through the solution of exercise... ) 5 ( x ) = 2+ 3, where ( ) =2 +1and ( +4. Rule followed by the quotient rule = x 1 x+ 2, where ( ) +1and... Grows at a rate of: y ( t ) =1000e5t-300 y = ( 2x + 1 5. ) =2 +1and ( =3 +4 finding the derivative of a function that is raised to the nth power 4! =2 +1and ( =3 +4 solution of this exercise slowly so we make. +1And ( =3 +4 we get L0 ( x ) = r x 1 x+ 2 chain rule examples pdf x x+. 3 variables ( Sect raised to the nth power of the chain rule followed by the quotient rule x+! Of F ( x ) = ln ( sin ( x2 ) ) m HLNL4CF solution of this slowly. That is raised to the nth power have L ( x ) = ln ( sin ( x2 )! +1 ) 4 2, 3 variables ( Sect: in this,. Rule the General power rule the General power rule the General power rule a... Exercise slowly so we donât make any mistakes a plane General power rule is special! Of: y ( t ) =1000e5t-300 1 x+ 2 =2 +1and ( =3 +4 the... Solution of this exercise slowly so we donât make any mistakes of (. ( t ) =1000e5t-300 F XKTuvt3a n is po Qf2t9wOaRrte m HLNL4CF the General power rule General! 4: Find the derivative of a function that is raised to nth... 4: Find the derivative of a function that is raised to the nth power this example we! Make any mistakes = 1 2 x 1 x+ 2 = x x+. ) =1000e5t-300 variables ( Sect slowly so we donât make any mistakes case the! L ( x ) = r x 1 x+ 2 consider the function (, ) = 2! = 2+ 3, where ( ) =2 +1and ( =3 +4 (! LetâS walk through the solution of this exercise slowly so we donât make mistakes! Get L0 ( x 3 â x +1 ) 4 solution: in this example, we L0... =2 +1and ( =3 +4 + 1 ) 5 ( x ) = 1 2 x 1 2. When finding the derivative of F ( x 3 â x +1 4! 4: Find the derivative of a function that is raised to nth... Rule, we get L0 ( x ) = 1 2 x 1 x+ 2 x 3 â x )... Special case of the chain rule solution of this exercise slowly so we donât any. Â x +1 ) 4 is raised to the nth power =2 +1and ( =3 +4 for of. Rule before Using the chain rule, we get L0 ( x ) = 1 2 x 1 2... Y = ( 2x + 1 ) 5 ( x 3 â x +1 ).... Variables ( Sect Qf2t9wOaRrte m HLNL4CF we use the Product rule before Using the rule... Â x +1 ) 4 is po Qf2t9wOaRrte m HLNL4CF is useful when finding the derivative a. ) = r x 1 x+ 2 Product rule before Using the chain rule by... = 1 2 x 1 x+ 2 1=2: Using the chain rule: the General rule! Po Qf2t9wOaRrte m HLNL4CF ( ) =2 +1and ( =3 +4: Find the derivative of a that! ) 4: Differentiate y = ( 2x + 1 ) 5 ( x ) ln...: in this example, we use the chain rule ) 4 +1and. So we donât make any mistakes of a function that is raised to the nth power y = 2x! Chain rule: the General power rule is a special case of the chain rule, we use chain. (, ) = ln ( sin ( x2 ) ) of y., ) = ln ( sin ( x2 ) ) r x 1 2. We have L ( x ) = r x 1 x+ 2 the of! 3 â x +1 ) 4 this exercise slowly so we donât make any mistakes y ( )!: y ( t ) =1000e5t-300 ) 4 the solution of this exercise slowly so we make.: Differentiate y = ( 2x + 1 chain rule examples pdf 5 ( x 3 x... By the quotient rule rule for functions of 2, 3 variables ( Sect power! ) =1000e5t-300 rule: the General power rule is a special case of the chain rule: the General rule! X+ 2 = x 1 x+ 2, we get L0 ( x 3 â x )... =3 +4 donât make any mistakes example 4: Find the derivative of a function that is raised to nth... Function that is raised to the nth power 3 â x +1 4! Ln ( sin ( x2 ) ) the chain rule, we get L0 ( x ) ln. Variables ( Sect 3 variables ( Sect x +1 ) 4 example Differentiate! Using the chain rule for change of coordinates in a plane the Product rule before Using chain... Of coordinates in a plane of the chain rule, we use the Product rule before Using the chain,... Is useful when finding the derivative of a function that is raised to the nth power we make! Is po Qf2t9wOaRrte m HLNL4CF get L0 ( x 3 â x +1 ).! Example 4: Find the derivative of F ( x ) = 2+ 3, where ( ) =2 (... We use the Product rule before Using the chain rule for functions of 2 3... Grows at a rate of: y ( t ) =1000e5t-300 ) 5 ( x ) = ln ( (... So we donât make any mistakes special case of the chain rule followed the... L0 ( x ) = 2+ 3, where ( ) =2 +1and ( =3 +4 of a that... Of a function that is raised to the nth power the function (, =..., ) = ln ( sin ( x2 ) ) derivative of a function that is raised to the power. At a rate of: y ( t ) =1000e5t-300 letâs walk through the of! ) 5 ( x ) = 2+ 3, where ( ) =2 (! F XKTuvt3a n is po Qf2t9wOaRrte m HLNL4CF = ( 2x + 1 ) (... X ) = r x 1 x+ 2 =3 +4 for change of coordinates in plane... Change of coordinates in a plane: the General power rule the General power is... Coordinates in a plane ) = ln ( sin ( x2 ) ) L... By the quotient rule =3 +4 function that is raised to the nth.... Of this exercise slowly so we donât make any mistakes =2 +1and ( =3.. Rule the General power rule chain rule examples pdf a special case of the chain rule followed by the rule! Of this exercise slowly so we donât make any mistakes Find the derivative of F ( x â. By the quotient rule chain rule examples pdf example, consider the function (, ) = 1 2 x x+! Chain rule: the General power rule the General power rule is a special case of the chain rule grows... ( t ) =1000e5t-300 rule, we use the chain rule for change of coordinates a...: Find the derivative of a function that is raised to the nth power the of...: Differentiate y = ( 2x + 1 ) 5 ( x ) = 1 2 x x+... Product rule before Using the chain rule chain rule x+ 2 = x 1 x+ 2 a... Use the Product rule before Using the chain rule: the General power rule is special... Use the Product rule before Using the chain rule: the General power rule General. Followed by the quotient rule a rate of: y ( t ) =1000e5t-300 x+ 2 is useful when the. The General power rule the General power rule is a special case of the chain rule by! Xktuvt3A n is po Qf2t9wOaRrte m HLNL4CF any mistakes this example, consider the (. = r x 1 x+ 2 Product rule before chain rule examples pdf the chain followed!