Notes for 12.4, 12.5

Notes for 12.4, 12.5 - ∂ Example 8 Chain Rule Two...

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Calculus III - Section 12.4 – Differentials We are going to extend the definition of differentials you learned in section 3.8 to a function of more than one variable. Differentials are used in their own right for error approximation, but you are going to get the “Cliff Notes” version of differentials. Definition of Total Differential If z f x y = ( , ) and x and y are increments of x and y , then the differentials of the independent variables x and y are dx x = ∆ and dy y = ∆ and the total differential of the dependent variable z is dz z x dx z y dy f x y dx f x y dy x y = + = + ( , ) ( , ) . This definition can be extended to functions if a greater number of variables in the usual way. Example: #4

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Calculus III – Section 12.5 – Chain Rules for Functions of Several Variables Chain Rule: One Independent Variable Let w f x y = ( , ) , where f is a differentiable function of x and y . If x=g(t) and y=h(t) , where g and h are differentiable functions of t , then w is a differentiable function of t , and dw dt w x dx dt w y dy dt = +
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Unformatted text preview: ∂ . Example: # 8 Chain Rule: Two Independent Variables Let w f x y = ( , ) , where f is a differentiable function of x and y . If x=g(s,t) and y=h(s,t) such that the first partials ∂ ∂ ∂ ∂ ∂ ∂ x s x t y s / , / , / , and ∂ ∂ y t / all exist, then ∂ ∂ w s / and ∂ ∂ w t / exist and are given by ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ w s w x x s w y y s and ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ w t w x x t w y y t . Example: #16 #20 Chain Rule: Implicit Differentiation If the equation F x y ( , ) = defines y implicitly as a differentiable function of x , then dy dx F x y F x y F x y x y y = -≠ ( , ) ( , ) , ( , ) . If the equation F x y z ( , , ) = defines z implicitly as a differentiable function of x and y , then ∂ ∂ = -∂ ∂ = -≠ z x F x y z F x y z z y F x y z F x y z F x y z x z y z z ( , , ) ( , , ) , ( , , ) ( , , ) , ( , , ) . Example: #28 #34...
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Notes for 12.4, 12.5 - ∂ Example 8 Chain Rule Two...

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