13) Give a function that requires three applications of the chain rule to differentiate. Now let = + − , then += (+ ). 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 … Proof of the Chain Rule • Given two functions f and g where g is diﬀerentiable at the point x and f is diﬀerentiable at the point g(x) = y, we want to compute the derivative of the composite function f(g(x)) at the point x. Be able to compare your answer with the direct method of computing the partial derivatives. Then lim →0 = ′ , so is continuous at 0. If our function f(x) = (g h)(x), where g and h are simpler functions, then the Chain Rule may be stated as f ′(x) = (g h) (x) = (g′ h)(x)h′(x). Proving the chain rule Given ′ and ′() exist, we want to find . Diﬀerentiation: Chain Rule The Chain Rule is used when we want to diﬀerentiate a function that may be regarded as a composition of one or more simpler functions. It is especially transparent using o() Guillaume de l'Hôpital, a French mathematician, also has traces of the Derivatives - Sum, Power, Product, Quotient, Chain Rules Name_____ ©W X2P0m1q7S xKYu\tfa[ mSTo]fJtTwYa[ryeD OLHLvCr._ ` eAHlblD HrgiIg_hetPsL freeWsWehrTvie]dN.-1-Differentiate each function with respect to x. In other words, we want to compute lim h→0 f(g(x+h))−f(g(x)) h. The Chain Rule is thought to have first originated from the German mathematician Gottfried W. Leibniz. Many answers: Ex y = (((2x + 1)5 + 2) 6 + 3) 7 dy dx = 7(((2x + 1)5 + 2) 6 + 3) 6 ⋅ 6((2x + 1)5 + 2) 5 ⋅ 5(2x + 1)4 ⋅ 2-2-Create your own worksheets like this … 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. Problems may contain constants a, b, and c. 1) f (x) = 3x5 2) f (x) = x 3) f (x) = x33 4) f (x) = -2x4 5) f (x) = - 1 4 (Section 3.6: Chain Rule) 3.6.2 We can think of y as a function of u, which, in turn, is a function of x. y=f(u) u=f(x) y=(2x+4)3 y=u3andu=2x+4 dy du =3u2 du dx =2 dy dx 21{1 Use the chain rule to nd the following derivatives. Let = +− for ≠0 and 0= ′ . Note that +− = holds for all . What if anything can we say about (f g)0(x), the derivative of the composition • The chain rule • Questions 2. The chain rule is the most important and powerful theorem about derivatives. Present your solution just like the solution in Example21.2.1(i.e., write the given function as a composition of two functions f and g, compute the quantities required on the right-hand side of the chain rule formula, and nally show the chain rule being applied to get the answer). Be able to compute the chain rule based on given values of partial derivatives rather than explicitly deﬁned functions. f0(u) = dy du = 3 and g0(x) = du dx = 2). Although the memoir it was first found in contained various mistakes, it is apparent that he used chain rule in order to differentiate a polynomial inside of a square root. Then differentiate the function. Chain Rule of Calculus •The chain rule states that derivative of f (g(x)) is f '(g(x)) ⋅g '(x) –It helps us differentiate composite functions •Note that sin(x2)is composite, but sin (x) ⋅x2 is not •sin (x²) is a composite function because it can be constructed as f (g(x)) for f (x)=sin(x)and g(x)=x² –Using the chain rule … Call these functions f and g, respectively. The Chain Rule Suppose we have two functions, y = f(u) and u = g(x), and we know that y changes at a rate 3 times as fast as u, and that u changes at a rate 2 times as fast as x (ie. 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