Find if y=sin(2 pi x+(pi/6)) for dy/dt. In the answers, I am solving it. Pls tell me my mistake. Also pls tell me some shortcut method to solve it.

We can use chain rule of differentiation. The chain rule of differentiation states that:
If z is a function of variable y, and y itself a function of variable x then we have dzdx=dzdy·dydx

In the given problem we have y is a function of x and x is a function of t. Now let us change the variable as 2πx+π6=z where z is also a function of t, then we have 2πdxdt=dzdt. So the given equation becomes y=sinz
differentiating with respect to t we get dydt=coszdzdt.
Now putting value of z we get dydt=cos2πx+π62πdxdt=2π cos2πx+π6dxdt. This is the one of the methods to solve this kind of problem.

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considering (2 pi x+(pi/6))=t d sin t/dx=dsint/dt × dt/dx=cos × 2 pi dx/dx+pid/6dx=cos × 2 pi. But actual answer is 2 pi cos(2 pi x+(pi/6)) But how. Pls solve in the same method so that I can know my mistake.
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