Question

Problem 1. In this problem, we'll explore another approach to implicit differ- entiation problems using the multivariable chain rule Suppose y is implicitly defined as a function of r by the equation F(, y)-0for some function F. For example, we could havez, уз-62y in plicitly defines y as a function of , in which case F(z, y)-2.3 + y3-62y (a) If F(x, y)0 mplicitly defines y as a function of , apply the chain rule to F(r, y) to show Hint: Differentiate F(,y) with respect to r using Chain Rule1, since y depends on implicitly and z depends on r by the equation , so that dr/dz = 1 .) (b) Use equation (1) to find dy/dr if y is an implicit function of z satisfying the equation co(y)-1 +sin(y Now, suppose z is implicitly defined as a function of a and y by the equation F(a,y,0 for some function F. For example, we could have y+ 3 +6xyz implicitly defines z as a function of and y, n which case (c) If F(z, y,z0 implicitly defines z as a function of r and y, apply the chain rule to F(z, y, z) to show 0: =-F,(z,y,z) = _OF/Oz and -=-F,(z.yiz) =-OF/0y Hint: Differentiation F(a,y,2) as in part (a) using Chain Rule 2, since z and y each depend on and y by the equations and y y with x/x -Oy/Oy-1 and z is implicitly dependent on z and y.) (d) Use equation (2) to find 0:/ and /y if z is an implicit function of r and y satisfying the equation yz + zlny

Answer 1

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