This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full Document
 Spring '12
 PamelaSatterfield
 Calculus, Approximation, Chain Rule, Derivative, differentiable function

Click to edit the document details