Question

Find the derivative of the function. \(y=3 x-5 \cos (2 x)^{2}\)

Short answer

The derivative of the function \(y = 3x - 5\cos(2x)^2\) is \(y'= 3 - 10 \cos(2x) \cdot -2\sin(2x)\) or \(y' = 3 + 20 \cos(2x) \sin(2x)\).

Step-by-step solution

Identify the composite function and apply the sums/difference rule

Identify the composite function in the original function \(y = 3x - 5\cos(2x)^2\) as \(\cos(2x)^2\), and the simple function as \(3x\). The derivative of a sum of functions is the sum of the derivatives, so the derivative of \(y\) can be determined by finding the derivative of each component separately.

Compute the derivative using constant multiple and chain rule

Calculate the derivative of the simple function \(3x\) directly and get 3. The derivative of the composite function \(-5\cos(2x)^2\) can be calculated by first identifying the outer function \(-5u^2\) (chain rule), where \(u = \cos(2x)\), and the inner function \(cos(2x)\). The derivative of \(-5u^2\) with respect to \(u\) is \(-10u\). By chain rule, multiply \(-10u\) by the derivative of the inner function to get the final derivative.

Compute the derivative of the inner function

The derivative of the inner function \(\cos(2x)\) could be worked out by using chain rule again. The outer function is \(\cos u\), where \(u = 2x\), and the inner function is \(2x\). The derivative of \(\cos u\) with respect to \(u\) is \(-\sin u\). By chain rule, multiply \(-\sin u\) by the derivative of the inner function to get the derivative of \(\cos(2x)\).

Assemble the full derivative

Lastly, combine the derivatives calculated in the previous steps. Substitute \(u = \cos(2x)\) back in to give the final derivative.