Derivative Rules

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Derivative of Cos(x)Sin(x)

Introduction

One of the fundamental concepts in calculus is the derivative, which measures the instantaneous rate of change of a function. In this article, we'll explore the derivative of the product of cosine and sine functions, cos(x)sin(x).

Derivation

To find the derivative of cos(x)sin(x), we'll use the product rule, which states that the derivative of a product of two functions f(x) and g(x) is given by: ``` (fg)'(x) = f'(x)g(x) + f(x)g'(x) ``` Applying this rule to cos(x)sin(x): ``` d/dx(cos(x)sin(x)) = d/dx(cos(x))sin(x) + cos(x)d/dx(sin(x)) ``` Using the power rule (derivative of x^a is ax^(a-1)): ``` = -sin(x)sin(x) + cos(x)cos(x) ``` Simplifying: ``` = -sin^2(x) + cos^2(x) ``` Using the trigonometric identity sin^2(x) + cos^2(x) = 1: ``` = 1 - sin^2(x) ``` Finally, the derivative of cos(x)sin(x) is: ``` d/dx(cos(x)sin(x)) = 1 - sin^2(x) ```

Conclusion

We have successfully derived the expression for the derivative of cos(x)sin(x), which is 1 - sin^2(x). This derivative provides valuable information about the instantaneous rate of change of the product function. By understanding the derivative, we can analyze the behavior of the function and gain insights into its characteristics.

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