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The chain rule. The chain rule is a powerful method used to differentiate composite functions (i.e. expressions where one function is contained in another function). An example of such a function would be. 2−3 +1, where the function 2 − 3 + 1 is contained inside the function .
1. Introduction. In this unit we learn how to differentiate a ‘function of a function’. We first explain what is meant by this term and then learn about the Chain Rule which is the technique used to perform the differentiation. 2. A function of a function. Consider the expression cos x2.
2 x - 9 y + x 3 y. 2 = sin ( y ) + 11 x . Remember y = y ( x ) here, so products/quotients of x and y will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y¢ (from the chain rule).
Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the function. Many answers: Ex y = ( ( ( 2 x + 1)5 + 2)6 + 3)7 dy = 7 ( ( ( 2 x + 1)5 + 2)6 + 3)6 ⋅ 6 ( ( 2 x + 1)5 + 2)5 ⋅ 5 ( 2 x + 1)4 ⋅ 2 dx.
Title: Calculus_Cheat_Sheet_All Author: ptdaw Created Date: 12/9/2022 7:12:41 AM
Derivatives Moderate Chain Rule 1. dx d cos 2x 2. dx d 2x +5 3. dx d 3 4. dx d sin x 5. dx d e 6. dx d ln x −5x 7. dx d 2x −1 8. dx d 4x−3 9. dx d 2x+6 10. dx d tan 2x
Summation Rule. a and k are constants. f, g, p, q, r, s, u, and v are functions of x such that f=f(x), g=g(x), p=p(x), q=q(x), r=r(x), s=s(x), u=u(x), and v=v(x), unless otherwise shown. (s(...))))) d/dx (∫ a f(t)dt) x. d/dx (∫a v(x) f(t)dt) d/dx (∫u(x) v(x) f(t)dt) d/dx (Σf(x)) f(x) f(v)v'.
