Understanding the Derivative of y sin x cos x with Respect to x
In calculus, one of the fundamental operations is differentiation. Specifically, the derivative of a function describes the rate of change of that function with respect to its input variable. In this article, we will explore the derivative of the function y sin x cos x with respect to x.
Introduction to the Problem
We are tasked with finding the derivative of the product y sin x cos x. This problem requires the use of the product rule, which is a fundamental technique in calculus for differentiating products of functions. The product rule states that for two functions u(x) and v(x), the derivative of their product is given by:
({d over dx} (u(x)v(x)) u'(x)v(x) u(x)v'(x))
Applying the Product Rule
Let's denote u(x) sin x and v(x) cos x. According to the product rule, we can find the derivative of y sin x cos x as follows:
[ frac{d}{dx} (sin x cos x) frac{d}{dx} (sin x) cdot cos x sin x cdot frac{d}{dx} (cos x) ]
Step-by-Step Calculation
First, we need to find the derivative of sin x: According to the derivative of the sine function, (frac{d}{dx} (sin x) cos x) Next, we need to find the derivative of cos x: According to the derivative of the cosine function, (frac{d}{dx} (cos x) -sin x)[ frac{d}{dx} (sin x cos x) (cos x) cdot cos x sin x cdot (-sin x) ]
Simplifying the Result
Now, we simplify the expression:
[ frac{d}{dx} (sin x cos x) cos^2 x - sin^2 x ]
However, this can be further simplified using the trigonometric identity:
[ cos^2 x - sin^2 x cos 2x ]
Conclusion
Thus, the derivative of the function y sin x cos x with respect to x is:
[ frac{d}{dx} (sin x cos x) cos 2x ]
Additional Insights and Applications
The derivative of (y sin x cos x) is not only important in theoretical calculus but also has practical applications. For instance, in physics, trigonometric functions describe periodic motion, and their derivatives can help in understanding the rates of change of such phenomena. Understanding the derivative of (sin x cos x) also aids in solving more complex problems involving product functions in calculus.