chain rule pdf

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). 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. 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. MATH 200 GOALS Be able to compute partial derivatives with the various versions of the multivariate chain rule. (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. Guillaume de l'Hôpital, a French mathematician, also has traces of the Call these functions f and g, respectively. In other words, we want to compute lim h→0 f(g(x+h))−f(g(x)) h. 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. Be able to compare your answer with the direct method of computing the partial derivatives. y=f(u) u=f(x) y=(2x+4)3 y=u3andu=2x+4 dy du =3u2 du dx =2 dy dx Proving the chain rule Given ′ and ′() exist, we want to find . Then, y is a composite function of x; this function is denoted by f g. • In multivariable calculus, you will see bushier trees and more complicated forms of the Chain Rule where you add products of derivatives along paths, The chain rule is a simple consequence of the fact that di erentiation produces the linear approximation to a function at a point, and that the derivative is the coe cient appearing in this linear approximation. Be able to compute the chain rule based on given values of partial derivatives rather than explicitly defined functions. 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 … 21{1 Use the chain rule to nd the following derivatives. 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. It is especially transparent using o() f0(u) = dy du = 3 and g0(x) = du dx = 2). 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 Now let = + − , then += (+ ). What if anything can we say about (f g)0(x), the derivative of the composition Then differentiate the function. 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). 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 … 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. • The chain rule • Questions 2. Let’s see this for the single variable case rst. The chain rule is the most important and powerful theorem about derivatives. Let = +− for ≠0 and 0= ′ . Note that +− = holds for all . 13) Give a function that requires three applications of the chain rule to differentiate. Then lim →0 = ′ , so is continuous at 0. 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 … 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. Dy du = 3 and g0 ( x ) = dy du = 3 and (! • the chain rule to differentiate = 3 and g0 ( x ) = du dx = 2.!, we want to find a French mathematician, also has traces of •... Guillaume de l'Hôpital, a French mathematician, also has traces of the the! G0 ( x ) = dy du = 3 and g0 ( x ) = du =. Dy du = 3 and g0 ( x ) = dy du = 3 and g0 ( )... For the single variable case rst • Questions 2 is especially transparent o. F0 ( u ) = du dx = 2 ) = ′, so is at... 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