The given equation is x*cos(y) - 2y = 0.
To find the derivative with respect to x, we will use the product rule and the chain rule (derivative of an outer function times derivative of the inner function).
Now, let's start differentiating.
First, we need to differentiate both sides of the equation with respect to x:
On the left-hand side, we have two terms, x*cos(y) and -2y, which we differentiate separately. The derivative of x*cos(y) can be computed using the product rule:
The derivative of the first term x is 1 and we leave the second term cos(y), plus we leave the first term x and find the derivative of the second term cos(y).
The derivative of cos(y) is -sin(y), but here y is a function of x. Therefore, we multiply -sin(y) by the derivative of the inside function by the chain rule, which is dy/dx.
So, the derivative of x*cos(y) with respect to x is: cos(y) - x*sin(y)*dy/dx
Next, we find the derivative of -2y with respect to x. As y is a function of x, we multiply -2 by the derivative of y with respect to x, which is dy/dx. Hence, the derivative of -2y with respect to x is: -2*dy/dx
On the right-hand side, we have zero, and the derivative of a constant is zero.
Putting it together, the first derivative of x*cos(y) - 2y with respect to x is: cos(y) - x*sin(y)*dy/dx -2*dy/dx
However, because we needed to find the first derivative of x with respect to x (which is one) and not y, we can simplify our derivative to : cos(y).
Again, to find the second derivative or d²y / dx², we would differentiate the equation again with respect to x. However, since the first derivative was a function of y (cos(y) ) which was constant with respect to x, derivative of a constant will be zero.
As a result, the second derivative d²y / dx² of the given equation is 0.
So, the first and the second derivatives of the given equation are cos(y) and 0 respectively.