Consider the following.
cos(x) + sqrt(y)= 1
(a) Find y' by implicit differentiation.
y' = 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 hope for a) you had
-sinx + (1/2)y^(-1/2) dy/dx = 0
dy/dx = 2√y sinx
b) from the original:
√y = 1 - cosx
square both sides
y = 1 - 2cosx + cos^2 x
dy/dx = 2sinx + 2cosx(-sinx)
= 2sinx(1 - cosx) , but 1-cosx = √y
= 2sinx √y , which matches my other dy/dx
To solve part (b), we start by rearranging the equation cos(x) + sqrt(y) = 1 to isolate sqrt(y):
sqrt(y) = 1 - cos(x)
To isolate y, we square both sides of the equation:
y = (1 - cos(x))^2
To find y', we differentiate both sides of the equation with respect to x:
y' = 2(1 - cos(x)) * (-sin(x))
Combining terms, we have:
y' = -2(1 - cos(x)) * sin(x)
Now let's move on to part (c). We'll substitute the expression for y from part (b) into the answer for y' from part (a) and see if they are consistent.
From part (b), we found that y = (1 - cos(x))^2. Now let's find y' using the answer from part (a), y' = 2(1 - cos(x)) * (-sin(x)):
y' = 2(1 - cos(x)) * (-sin(x))
= -2(1 - cos(x)) * sin(x)
Now we substitute y = (1 - cos(x))^2 into y':
y' = -2(1 - cos(x)) * sin(x)
= -2((1 - cos(x))^2) * sin(x)
By comparing the expression for y' in part (a) and the substituted expression in part (c), we can see that they are indeed consistent.
(b) To solve the equation explicitly for y, we need to isolate y on one side of the equation.
Starting with the equation: cos(x) + sqrt(y) = 1
Subtracting cos(x) from both sides: sqrt(y) = 1 - cos(x)
Squaring both sides: y = (1 - cos(x))^2
To find y' in terms of x, we can differentiate this equation with respect to x using the chain rule:
dy/dx = 2(1 - cos(x)) * (-sin(x))
So, y' = -2(1 - cos(x)) * sin(x)
(c) To check the consistency of the solutions, we'll substitute the expression for y from part (b) into the solution for part (a).
In part (a), we found that y' = -sin(x) / (2 sqrt(y))
Substituting y = (1 - cos(x))^2 into y', we get:
y' = -sin(x) / (2 sqrt((1 - cos(x))^2))
Simplifying the expression, we have:
y' = -sin(x) / (2 |1 - cos(x)|)
Since we know that y = (1 - cos(x))^2 is always positive, we can simplify the absolute value to remove it:
y' = -sin(x) / (2 (1 - cos(x)))
Therefore, the solutions from part (a) and part (b) are consistent.