To find the derivative of the function y^3 * ln(x^5 ) - 5x^2 - y^4 with respect to x and y, we will use the concept of partial derivatives.
First, we perform a partial derivative with respect to x.
Remember that the derivative of ln(u) is 1/u and the derivative of u with respect to x is du/dx. Here, we treat y as a constant and apply the chain rule for the function y^3 * ln(x^5).
So the derivative of y^3 * ln(x^5 ) with respect to x giving us the result: 5*y^3/x. The derivative of -5x^2 with respect to x is -10x and the derivative of - y^4 with respect to x is 0 because y is treated as a constant.
Then summing these three parts together gives us -10x + 5*y^3/x.
Second, we perform a partial derivative with respect to y. Here, x is a constant and for the function y^3 * ln(x^5) the power rule gives : 3*y^2 * ln(x^5) and the derivative of y^4 becomes -4*y^3.
So, the result for derivative with respect to y is: -4*y^3 + 3*y^2*log(x^5).
To find dx/dy, we divide the derivative with respect to y by the derivative with respect to x:
dx/dy = (-4*y^3 + 3*y^2*log(x^5)) / (-10*x + 5*y^3/x),
this is our dx/dy value.
To summarize the answer: the derivative with respect to x is -10*x + 5*y^3/x, the derivative with respect to y is -4*y^3 + 3*y^2*log(x^5), and dx/dy equals (-4*y^3 + 3*y^2*log(x^5)) / (-10*x + 5*y^3/x).