What is the differential of y =sin(2x)?

For this question we have to use the chain rule. The chain rule is dy/dx (the differential you are looking for) = du/dx (the differential of u with respect to x) * dy/du(the differential of y with respect to u). The The reason this works is because the when you multiply du/dy and dx/du, the du terms cancel out leaving just dy/dx.

To begin to find the differential of y = sin(2x) we need to use a substitution. In this case we use u = 2x. Then we substitute the u into the original question resulting in y = sin(u). We do this as it is simple for us to differentiate sin(u). Once we make the substitution we have two equations to use. Firstly y = sin(u) and secondly u = 2x. Now we will differentiate both equations. When we differentiate u = 2x, the result is du/dx = 2. Then once we differentiate y = sin(u) we end up with dy/du = cos(u). As we now have dy/du and du/dx we can substitute them into the chain rule formula which results in dy/dx = (2)*(cos(u)). Now to finish we need to remove the u term. This is simple as we know that u = 2x and therefore dy/dx = 2cos(2x).