EXERCISE 8.4

EXERCISE . . Find the partial derivatives of the following functions at the indicated points. (i) f x y ( , (ii) g x y ( , (iii) h x y z z x ( , , ) sin( (iv) G x y log( ),( , ) + . For each of the following functions find the f , and show that f yx (i) f x y (ii) f x y tan − (iii) f x y cos( . If U x y z z y ( , , ) = , find ∂ U U , and ∂ U z . . If U x y z z ( , , ) log( , find U U U z . . For each of the following functions find the g g g xx yy and g yx . (i) g x y xe x y ( , ) = + (ii) g x y log( (iii) g x y cos( . Let w x y z z x y z ( , , ) , ( , , ) ( , , ) ≠ . Show that ∂ + ∂ + ∂ w w w z . If V x y ( cos sin ) , then prove that ∂ + ∂ V V . If w x y sin( , then prove that w y x w x y . and and . If v x y z z xyz ( , , ) = , show that ∂ ∂∂ = ∂ ∂∂ v y z v z y . . A firm produces two types of calculators each week, x number of type A and y number of type B . The weekly revenue and cost functions (in rupees) are R x y and C x y ( , ) = respectively. (i) Find the profit function P x y ( , ) , (ii) Find ∂ P x ( and ∂ P y ( and interpret these results.