Find the differential dy of the given function
y= (1/3) cos((6pi times x-1))/((2))
Can anyone help me with this problem?
Do you know the chain rule? But write:
$\displaystyle y=\frac{1}{3}\cdot \left[\frac{\cos(6\pi (x-1))}{2}\right]=\frac{1}{6}\cdot \left[\cos(6\pi (x-1))\right]$
But it's still not clear, you have to be more clear with: $\displaystyle \cos(6\pi(x-1))$ or $\displaystyle \cos(6\pi\cdot x-1)$?
If you want to use the chain rule here:
$\displaystyle D\left[\cos\left(\frac{6\pi x-1}{2}\right)\right]=-\sin\left(\frac{6\pi x-1}{2}\right)\cdot D\left(\frac{6\pi x-1}{2}\right)$. Can you go further? You don't have to use the quotienrule, just put $\displaystyle \frac{1}{2}$ outside, because it's a constant.
$\displaystyle y = \frac{1}{3} \cos\left(3\pi x - \frac{1}{2}\right)$
$\displaystyle \frac{dy}{dx} = -\frac{1}{3} \sin\left(3\pi x - \frac{1}{2}\right) \cdot 3\pi$
$\displaystyle \frac{dy}{dx} = -\pi \sin\left(3\pi x - \frac{1}{2}\right)$