Question

Find the differential \(d y\) of the given function. $$ y=\frac{1}{3} \cos \left(\frac{6 \pi x-1}{2}\right) $$

Short answer

The differential of the given function is \( -\pi \sin \left( \frac{6 \pi x - 1}{2} \right) \)

Step-by-step solution

Identify Function Type

First, identify that the function is a composite function, which means a function is being applied to another function. The inner function is \(\frac{6 \pi x - 1}{2}\) and the outer function is \(\frac{1}{3} \cos x\).

Apply the Chain Rule

Next, apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Differentiate the outer function with respect to its inner function to give: \( -\frac{1}{3} \sin \left( \frac{6 \pi x - 1}{2} \right) \). Then differentiate the inner function with respect to \(x\) obtaining \(3\pi\) .

Multiply the results

Now, multiply the results from the differentiation of outer function and the inner function. This gives the differential \(d y\) of the given function: \( -3\pi \sin \left( \frac{6 \pi x - 1}{2} \right) /3 \).