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Okay, so we're given the function *y = sin(x)cos(x)*. Our mission? To find its derivative, which we denote as *dy/dx*. The derivative tells us the instantaneous rate of change of the function at any given point. In simpler terms, it describes the slope of the tangent line at any point on the curve. This is not just a mathematical exercise; it's a fundamental concept in physics, engineering, and many other fields. The product of sine and cosine is a common expression in trigonometric functions, and knowing how to deal with this kind of expression is essential. This particular function blends the characteristics of both sine and cosine waves, so the derivative will show us how these waves interact and change together. Essentially, we want to figure out how *y* changes as *x* changes. The *dy/dx* notation is just a concise way of representing this relationship. We can't just look at *sin(x)* and *cos(x)* separately and say, "Hey, the derivative of that is..." because they're multiplied together. This is where the product rule comes to the rescue! We have to treat this as a product of two functions.
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