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 $\frac{dz}{dx}=\frac{dz}{dy}·\frac{dy}{dx}$

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 $2\mathrm{\pi x}+\frac{\mathrm{\pi }}{6}=\mathrm{z}$ where z is also a function of t, then we have $2\pi \frac{dx}{dt}=\frac{\mathrm{dz}}{\mathrm{dt}}$. So the given equation becomes $y=\mathrm{sin}\left(z\right)$
differentiating with respect to t we get $\frac{dy}{dt}=\mathrm{cos}\left(z\right)\frac{dz}{dt}$.
Now putting value of z we get . 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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