The curve C has equation: 2(x^2)y + 2x + 4y – cos(pi*y) = 17. Use implicit differentiation to find dy/dx in terms of x and y.
The first step is to differentiate both sides of this equation with respect to x - we will then be able to solve for dy/dx. Differentiating the right side of the equation gives d/dx(17)=0. We’ll differentiate the left side term by term. Term 1: d/dx(2x2y). We use the product rule to give yd/dx(2x2)+2x2(dy/dx)=4xy+2x2(dy/dx). Term 2: d/dx(2x)=2. Term 3: d/dx(4y). We use the chain rule, du/dx=(du/dy)(dy/dx), with u=4y. This gives d/dx(4y)=d/dy(4y)dy/dx=4(dy/dx). Term 3: d/dx(-cos(piy)). We use the chain rule, with u=-cos(piy). This gives d/dx(-cos(piy))=d/dy(-cos(piy))(dy/dx). To find d/dy(-cos(piy)), we use the chain rule once more, with u=piy, to give d/dy(-cos(u))=d/du(-cos(u))(du/dy)=pisin(u)=pisin(piy). Therefore, d/dx(-cos(piy))=pisin(piy)(dy/dx). The differential of cos is a standard result that should be remembered - it is not expected for it to be derived in this question. Now we have differentiated all the terms and the resultant expression is: 4xy+2x2(dy/dx)+2+4(dy/dx)+pisin(piy)(dy/dx)=0. Finally, we rearrange this expression to find dy/dx in terms of x and y: dy/dx=(-4xy-2)/(2x2+4+pisin(pi*y)).
