Determine dy/dx if xsiny=artany
i did (x)(cosy dy/dx) so far
To find dy/dx in the equation xsiny = artany, you can use implicit differentiation. Here's how:
1. Start with the given equation: xsiny = artany.
2. Take the derivative of both sides of the equation with respect to x. Remember, when taking the derivative of y with respect to x, you need to apply the Chain Rule.
3. For the left-hand side, the derivative of xsiny with respect to x requires both the product rule and the chain rule.
a. The derivative of x with respect to x is 1.
b. The derivative of sin(y) with respect to x is cos(y) * dy/dx.
So, the left-hand side becomes: 1 * siny + x * (cos(y) * dy/dx).
4. For the right-hand side, the derivative of artany with respect to x also requires the chain rule.
a. The derivative of artan(y) with respect to y is 1 / (1 + y^2).
b. The derivative of y with respect to x is dy/dx.
So, the right-hand side becomes: (1 / (1 + y^2)) * dy/dx.
5. Now, set the left-hand side equal to the right-hand side and solve for dy/dx:
1 * siny + x * (cos(y) * dy/dx) = (1 / (1 + y^2)) * dy/dx.
6. Simplify the equation by moving all terms involving dy/dx to one side:
x * (cos(y) * dy/dx) - (1 / (1 + y^2)) * dy/dx = -siny.
7. Factor out dy/dx from the left-hand side:
dy/dx * (x * cos(y) - 1 / (1 + y^2)) = -siny.
8. Finally, solve for dy/dx by dividing both sides by (x * cos(y) - 1 / (1 + y^2)):
dy/dx = -siny / (x * cos(y) - 1 / (1 + y^2)).
So, dy/dx is equal to -siny / (x * cos(y) - 1 / (1 + y^2)).