use implicit differentiation to solve
dy/dr sqrt(r+y) = sin(r)cos(y)
To solve this equation using implicit differentiation, we will first take the derivative of both sides of the equation with respect to r.
For the left side, we can use the chain rule. Let's call the left side of the equation f(r, y) = √(r + y). Taking the derivative with respect to r, we get:
df/dr = (1/2)(r + y)^(-1/2) * (1 + dy/dr)
For the right side, we can simply take the derivative of sin(r)cos(y) with respect to r, which is -sin(r)sin(y).
Now, our equation becomes:
(1/2)(r + y)^(-1/2) * (1 + dy/dr) = -sin(r)sin(y)
Next, we want to solve for dy/dr. We can rearrange the equation as follows:
(r + y)^(-1/2) * (1 + dy/dr) = -2sin(r)sin(y)
Multiply both sides by (r + y)^(1/2):
1 + dy/dr = -2sin(r)sin(y)(r + y)^(1/2)
Now, isolate dy/dr:
dy/dr = -2sin(r)sin(y)(r + y)^(1/2) - 1
Therefore, dy/dr = -2sin(r)sin(y)(r + y)^(1/2) - 1 is the solution using implicit differentiation.