Consider the following.
cos(x) + sqrt(y)= 1
(a) Find y' by implicit differentiation.
y' = 2y^(1/2) sin(x)
Correct: Your answer is correct.
(b) Solve the equation explicitly for y and differentiate to get y' in terms of x.
y' = ?
(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a).
y' =?
I did this for you here:
https://www.jiskha.com/display.cgi?id=1526060796
Why are you reposting it ?
It wasn't correct and I wanted to post my answer I got for context
But it is correct.
Proof:
http://www.wolframalpha.com/input/?i=find+dy%2Fdx+for+cos(x)+%2B+sqrt(y)%3D+1
cos(x) + sqrt(y)= 1
-sin x dx/dx + (1/2) y^-.5 dy/dx = 0
dy/dx = 2 sin x /y^-.5 = 2 sin x sqrt y
so it is surely correct
cos(x) + sqrt(y)= 1
sqrt y = 1 - cos x
y = 1 -2 cos x + cos^2 x
dy/dx = 2 sin x - 2 cos x sin x = 2 sin x (1- cos x) That is part 2
but we know that 1 - cos x = sqrt y
so
dy/dx = 2 sin x sqrt y (again)
that shows they are consistent
To solve part (b), we need to explicitly solve the equation for y and then differentiate to find y' in terms of x.
Starting with the equation:
cos(x) + sqrt(y) = 1
We can isolate the square root term:
sqrt(y) = 1 - cos(x)
To eliminate the square root, we square both sides of the equation:
y = (1 - cos(x))^2
Now, we can differentiate both sides with respect to x to find y':
Using the chain rule, the derivative of y with respect to x is:
y' = 2(1 - cos(x)) * (-sin(x))
Simplifying this expression, we get:
y' = -2(1 - cos(x))sin(x)
So, the answer to part (b) is:
y' = -2(1 - cos(x))sin(x)
Moving on to part (c), we can substitute the expression for y in the solution from part (b) into the solution from part (a) to check for consistency.
From part (a), we found that:
y' = 2y^(1/2) * sin(x)
Substituting y = (1 - cos(x))^2 into this equation:
y' = 2(1 - cos(x))^((1/2)) * sin(x)
And simplifying this expression gives:
y' = 2(1 - cos(x))^(1/2) * sin(x)
Which matches the answer we obtained in part (b). Therefore, the solutions from parts (a) and (b) are consistent.
So, the answer to part (c) is:
y' = 2(1 - cos(x))^(1/2) * sin(x)