Implicit Differentiation Question Help?

Question

Consider the following:

cos(x)+#sqrt(y)# = 5

A) Find y' by implicit differentiation

B) Solve the equation explicitly for y and differentiate to get y' in terms of x.

C) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a).

Help!?

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Answer 1 · 2018-02-13T06:41:05

Please refer to the Explanation.

Part A) :

#cosx+sqrty=5#.

Diff.ing w.r.t. #x, d/dxcosx+d/dxsqrty=d/dx5#.

#:. -sinx+d/dy(sqrty)*dy/dx=0...[because," the Chain Rule]"#.

#:. -sinx+1/(2sqrty)*dy/dx=0#.

#:. y'=dy/dx=2sqrty*sinx#.

Part B) :

#cosx+sqrty=5#.

#:. sqrty=5-cosx#.

#"Squaring, "y=(5-cosx)^2#.

#:. dy/dx=d/dx(5-cosx)^2,#

#=2(5-cosx)*d/dx(5-cosx).......[because," the Chain Rule]"#,

#=2(5-cosx)*{0-(-sinx)}#.

#rArr y'=2sinx(5-cosx)#.

Part C) :

As in Part A), #y'=2sqrty*sinx#.

But, given that, #sqrty=5-cosx#.

Sub.ing, we find, #y'=2(5-cosx)sinx=2sinx(5-cosx)#,

which shows that the solutions derived in Parts A) and B)

are perfectly matching!

Hence, the verification.