Question

In Exercises \(13-24,\) find \(d y / d x .\) If you are unsure of your answer, use NDER to support your computation. $$y=\sin ^{2}(3 x-2)$$

Short answer

The derivative \(dy/dx\) of the function given is \(6\sin(3x - 2)\cos(3x - 2)\).

Step-by-step solution

Identify the Inner Function

Start by identifying the inner function. In the given equation, the inner function is \(3x - 2\).

Differentiate the Outside Function

Differentiate the outside function, which is the sine function squared. However, keep the inside function as it is. So, the derivative of \(\sin^2 (u)\) with respect to \(u\) is \(2\sin(u)cos(u)\). Hence the derivative of the outside function is \(2 \sin(3x - 2)\cdot \cos(3x - 2)\).

Apply the Chain Rule

Now, apply the chain rule. When we differentiate the inner function \(u = 3x - 2\) with respect to \(x\), we get \(du/dx = 3\). Multiply the derivative of the outer function by the derivative of the inner function to get the derivative of the entire function. So, \(dy/dx = 3 \cdot 2 \sin(3x - 2) \cdot \cos(3x - 2)\).

Simplify the Result

Finally, simplify the result to get the final solution. The result is \(dy/dx = 6\sin(3x - 2)\cos(3x - 2)\).