Understanding the Derivative of sin(cos x) with Respect to x

In the realm of calculus, determining the derivative of a composite function such as sin(cos x) is a fundamental task. This article provides a comprehensive guide to understanding the process and significance of finding the derivative of sin(cos x) with respect to x. We will explore the chain rule and the product rule, and delve into detailed step-by-step derivations.

The Chain Rule in Action

The chain rule is a powerful tool in calculus that allows us to differentiate composite functions. For the function y sin(cos x), we can apply the chain rule as follows:

Step-by-Step Derivation Using the Chain Rule

Identify the Inner and Outer Functions: In the function y sin(cos x), the inner function is u cos x, and the outer function is sin u. Differentiate the Outer Function: The derivative of the outer function sin u with respect to u is cos u. Differentiate the Inner Function: The derivative of the inner function u cos x with respect to x is -sin x. Apply the Chain Rule: dy/dx (d/dx sin u) * (d/dx u) cos u * -sin x. Substitute u cos x: This yields the final derivative as dy/dx cos(cos x) * -sin x, or -sin x * cos(cos x).

This step-by-step application of the chain rule illustrates how to differentiate the function sin(cos x) with respect to x. The process can be summarized as:

d/dx [sin(cos x)] cos(cos x) * -sin x.

The Product Rule: A Deeper Dive

While the chain rule provides a straightforward approach, sometimes the problem at hand may require the use of the product rule. The product rule is used to differentiate the product of two functions, such as:

d/dx [sin x * cos(ax)] cos x * cos(ax) - sin x * sin(ax).

Apply the Product Rule: The product rule states that if we have a function uv, where u and v are both functions of x, then the derivative is given by d/dx [uv] u * dv/dx v * du/dx. Differentiate Each Function: For u sin x: The derivative is cos x. For v cos(ax): The derivative is -a sin(ax). Combine the Results: Plugging these derivatives back into the product rule formula, we get d/dx [sin x * cos(ax)] sin x * (-a sin(ax)) cos x * cos(ax).

This method provides an alternative approach to finding the derivative and demonstrates the flexibility of calculus in different contexts.

Conclusion

Understanding the derivative of sin(cos x) with respect to x is crucial for advanced calculus and mathematical analysis. Whether using the chain rule, the product rule, or other techniques, these methods provide a robust framework for solving complex differentiation problems. By mastering these techniques, you can efficiently handle a wide range of mathematical challenges involving composite functions.

Keywords: derivative of sin(cos x), chain rule, product rule