In this problem, we're given this equation, 4times(sin-1(x))+(cos-1(y))=π3. Now we want to find dy/dx at the ordered pair (0, 1/2). So, in order to do that, we're going to have to use implicit differentiation on this equation here and take ddx of each side. So this is going to be a ddx of 4times(sin-1(x))+(cos-1(y)). And this is then equal to ddx, my derivative of π3.
So now we need to take these derivatives with our ultimate goal to find dy/dx. Now starting with the easiest side, I know that the derivative of this constant is just 0. So taking it back to our left side here, the derivative of 4times(sin-1(x)) is just going to be 4/1-(x2). Then the inverse cosine of y taking this derivative is a bit more complicated because this is y and not x. We're adding this together with -1/1-(y2), then applying the chain rule here.
This is times dy/dx, which is ultimately what we want to find here. So our goal from this point is just to get this dy/dx by itself. So I'm going to go ahead and subtract this over to the other side. So this becomes -1/1-(y2) times dy/dx is then equal to -4, since I'm subtracting this over to the other side, over the square root of 1 minus x squared. Then from here, I can go ahead and multiply by the square root of 1 minus y squared.
And I can go ahead and cancel these negatives on either side, since I have a negative on each side, they cancel each other out. So this multiplication by the square root of 1 minus y squared will cancel over here, and I'm just left with dy/dx. And then this is equal to 4times1-(y2) over the square root of 1 minus x squared. But here, we were given values for x and y.
So that means that from here, we can actually plug these values in. We were given that y is equal to 0 and x is equal to 1/2. So that's what we want to plug in and now that we're at this final step. So this is 4times1-(122) over the square root of 1 minus 0 squared. Now on the bottom, that zero goes away.
We're just left with the square root of 1. That's just 1. Now up here in my top square root in my numerator, this then becomes 4times1-(122). That is 1 minus 1 fourth. 1 minus 1 fourth is 3 fourths, so this is the square root of 3 fourths.
Now we can simplify that further because the square root of 3 fourths is root 3 over 2. So this is then equal to 4root3/2. And further simplifying that, 4 over 2 just gives me 2. So this has been 2 root 3. For my final answer here, dy/dx at the ordered pair (0, 1/2).
Let us know if you have any questions here, and let's keep going.