Find Second Derivative of x^2 + y^2 = 1
How to find second derivative of x^2 + y^2 = 1
we have a second derivative using implicit differentiation the first derivative is fairly easy to find here. We're gonna same thing as before we're going to do the derivative but then also finish whatever term. We have with either DX DX or dy DX depending on what we were working with the first term is 2x we finished with DX DX because it's a X variable that we're working with that we're doing the derivative of and then. We add 2y alright and then we finish that term with a dy/dx because that function has a Y in it all right and the other side is going to equal zero and this is important that. we're going to solve for dy/dx that's the first step we're going to let this go to one that goes away we're going to subtract 2x from both sides.
We would get dy dy DX equals negative 2x and then we're going to divide both sides by 2y so we get dy DX equals negative 2x over 2y or just negative x over Y and then we're going to treat this as a quotient rule we're going to take the derivative of both sides but we're going to be careful to remember that dy/dx and negative x over Y are interchangeable so when we do the derivative of this the second derivative we get d squared y DX squared on the left side on the right side we're going to run through the quotient rule we're gonna bring the top up minus y times 1 DX over DX right still using the implicit differentiation rules of finishing and then minus x times 1 dy DX and then that's all over Y squared which is the original denominator here this DX DX goes to 1 right but remember I said up here right is that dy DX's can now be replaced with negative x over y so when we clean up this we get negative y minus x dy/dx all over y squared all right, but when we replace the dy/dx with negative x over y we end up with what's the y minus x times x over y all over y squared we get y plus x squared over Y over Y squared mmm to get a common denominator we get Y squared over Y because we need a Y in the bottom and then plus x squared over Y all over Y squared okay and here we're going to go all the way back to the original alright we know that x squared plus y squared is equal to one so when we start to simplify this fraction which is still d squared Y over DX squared we put together these two fractions on the top we end up with y squared plus x squared all over why there's minus sign missing from here and then this is all over Y squared so this x squared plus y squared is equal to 1 right and that's from the original statement and so we can get 1 over Y negative over Y squared right and if we simplify this we get negative 1 over Y times 1 over Y squared right you flip a second fraction and multiply by one and that's. how we end up with negative 1 over y cubed and it all works out.