Derivative of Secant Function 4x

Find the derivative of $\sec(4x)$.

When faced with the task of finding the derivative of the secant function, particularly in the form $\sec(4x)$, it is essential to recall some underlying principles of calculus, especially those concerning trigonometric functions and their derivatives. The derivative of the secant function, $\sec(x)$, is a fundamental component in this process. To solve for derivatives of composite trigonometric functions, the chain rule becomes a powerful tool. The chain rule states that to differentiate a composite function, you must take the derivative of the outer function and multiply it by the derivative of the inner function.

For the specific case of $\sec(4x)$, you need to differentiate the outer function, $\sec(u)$, with respect to its argument, $u$, and then multiply it by the derivative of the inner function, $4x$, with respect to $x$. This systematic approach is crucial when dealing with more complex functions and is a recurring theme in differential calculus. Understanding this strategy not only helps in handling trigonometric derivatives but also builds a strong foundation for more advanced calculus topics, such as inverse trigonometric functions and differentiation of more intricate composite functions.

Moreover, recognizing standard derivatives of trigonometric functions can expedite the differentiation process. For example, the derivative of $\sec(x)$ with respect to $x$ is $\sec(x) \tan(x)$. Such fundamental identities are key to tackling calculus problems efficiently. Overall, mastering these strategies assists in navigating other mathematical contexts where trigonometric functions are employed, be it in physics, engineering, or further mathematical studies.

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