... if only there was some special way you could search for a list of derivatives with your computer.i have specially doubt on the partial derivative of U w.r.t y for inverse sin.
A partial derivative is a mathematical concept used in multivariable calculus to calculate the rate of change of a function with respect to one of its variables while holding all other variables constant.
To find the partial derivative of a function, you take the derivative with respect to the variable in question while treating all other variables as constants. In the case of the function U = sin^-1(x/y) + cos(y/x), we would take the partial derivative with respect to x and y separately.
The partial derivative of U = sin^-1(x/y) with respect to x is 1/√(1-(x/y)^2). This can be found by using the chain rule and the derivative of the inverse sine function.
The partial derivative of U = cos(y/x) with respect to y is -1/x*sin(y/x). This can be found by using the chain rule and the derivative of the cosine function.
The partial derivative of U = sin^-1(x/y) + cos(y/x) is 1/√(1-(x/y)^2) - 1/x*sin(y/x). This can be found by taking the partial derivatives of each term and adding them together.