find the derivative of xy^3 cos x + x^2y = 1 with step by step working out

To find the derivative of the given function, we will use the product rule and the chain rule. The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

The function is: f(x, y) = xy^3 cos x + x^2y - 1

Step 1: Differentiate each term separately.

The derivative of xy^3 cos x with respect to x is:

d/dx[xy^3 cos x] = y^3 cos x + xy^3 (-sin x) (using the product rule)

The derivative of x^2y with respect to x is:

d/dx[x^2y] = 2xy + x^2 dy/dx (using the product rule)

The derivative of 1 with respect to x is 0.

Step 2: Add up all the derivatives.

d/dx[f(x, y)] = y^3 cos x - xy^3 sin x + 2xy + x^2 dy/dx

Step 3: Rearrange the equation to solve for dy/dx.

dy/dx = (y^3 cos x - xy^3 sin x + 2xy) / x^2

So, the derivative of the given function is dy/dx = (y^3 cos x - xy^3 sin x + 2xy) / x^2.