Final answer:
The derivative of 1-y=cos(xy) with respect to x is found using implicit differentiation and chain rule, resulting in an expression for dy/dx.
Step-by-step explanation:
The problem asks us to find the derivative of the equation 1 - y = cos(xy) with respect to x. This involves understanding implicit differentiation because y is a function of x. Let's proceed with the differentiation step by step:
- First, we differentiate both sides of the equation with respect to x, remembering that y is a function of x, which makes it necessary to use the chain rule.
- The derivative of 1 with respect to x is 0, and the derivative of -y with respect to x is -dy/dx because y is a function of x.
- To differentiate cos(xy) with respect to x, we apply the chain rule: the derivative of cos is -sin, but since the argument is xy, we multiply by the derivative of xy, which is y + x(dy/dx).
- Equating the derivatives of both sides gives us 0 - dy/dx = -sin(xy) · (y + x(dy/dx)). To solve for dy/dx, we bring all terms involving dy/dx to one side and factor it out.
- The final form after we solve for dy/dx is the derivative of y with respect to x in terms of x and y.
To summarize, we have used implicit differentiation and the chain rule to differentiate a trigonometric function of a product of variables.