Question

True or False? In Exercises 137-139, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=\sin ^{2}(2 x),\) then \(f^{\prime}(x)=2(\sin 2 x)(\cos 2 x)\)

Short answer

The statement is false. The correct derivative of the function \(f(x) = \sin^{2}(2x)\) is \(f'(x) = 4 \sin(2x) \cos(2x)\), not \(2(\sin 2x)(\cos 2x)\) as stated.

Step-by-step solution

Identify the function in the statement

The function in the statement is \(f(x) = \sin^{2}(2x)\).

Apply the chain rule to differentiate the function

The chain rule states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. Here, the outer function is \(\sin^{2}(x)\) and the inner function is \(2x\). Consequently, the derivative of \(f(x)\) is \(f'(x) = 2 \sin(2x) \cdot \cos(2x)\cdot2.\)

Compare the calculated derivative with the statement

The calculated derivative is \(4 \sin(2x) \cos(2x)\), not \(2(\sin 2x)(\cos 2x)\). This shows that the statement is false.

Provide an explanation to complete the problem

The statement is false because the correct derivative of \(f(x) = \sin^{2}(2x)\) is \(f'(x) = 4 \sin(2x) \cos(2x)\), not \(2(\sin 2x)(\cos 2x)\) as stated in the exercise.