Is there an easy explanation of implicit differentiation? Where do I start to solve y" given
1) x+xy=y=2
2) x^3-3xy+y^3
Please help!!
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/ImplicitDiff.html
Take the first one (read the examples in the link above)
x + xy + y =2
take the derivative of all with respect to x.
1 + y + xy'+y'=0
solve this in terms of y'
y'= - (y+1)/(x+1)
How do I find the point (x,y) in the first quadrant that lies on the curve x^3+y^3=3xy and has the largest possible y-coordinate
To find the point (x, y) in the first quadrant that lies on the curve x^3+y^3=3xy and has the largest possible y-coordinate, you need to use the concept of optimization.
Here's how you can approach this problem:
1. Start by differentiating the equation x^3+y^3=3xy implicitly with respect to x. This will give you dy/dx in terms of x and y.
2. Simplify the derivative obtained in step 1 by solving for dy/dx.
3. Since you want to find the point with the largest possible y-coordinate, you need to find the critical points where dy/dx equals zero.
4. Set dy/dx equal to zero and solve for x. This will give you the x-coordinate(s) of the critical point(s).
5. To determine which critical point(s) lies in the first quadrant, substitute the x-coordinate(s) obtained in step 4 back into the original equation x^3+y^3=3xy and solve for y.
6. Compare the y-coordinates obtained in step 5 and choose the one with the largest value.
By following these steps, you can find the point (x, y) that satisfies the given conditions.