Firstly, let us identify the given function which is sinh(x^2 * y) - arcsin(y+x) + 10. To find its derivative, we need to apply the chain rule and implicit differentiation. Let's begin:
1. Look at the first term of the above equation, which is sinh(x^2 * y). The derivative of sinh(u) with respect to u is cosh(u). Therefore, by the chain rule, the derivative of sinh(x^2 * y) with respect to x is cosh(x^2 * y) * derivative of (x^2 * y) with respect to x. Further applying the chain rule, we will get the derivative of x^2 * y with respect to x is 2 * x * y. Therefore, the overall derivative of the first term is 2*x*y*cosh(x^2*y).
2. The second term to differentiate is -arcsin(y + x). The derivative of arcsin(u) with respect to u is 1/sqrt(1 - u^2). Hence, using the chain rule, the derivative of -arcsin(y+x) with respect to x is -1/sqrt(1 - (y+x)^2).
3. The last term is a constant, 10, and the derivative of a constant is zero.
4. Summing up the results from steps 1 to 3, the derivative of the whole function, dy/dx, is 2*x*y*cosh(x^2*y) - 1/sqrt(1 - (x + y)^2).