91Ó°ÊÓ

Trigonometric functions, such as sine, cosine, and tangent, have related functions called cosecant, secant, and cotangent. The differentiation of these functions follows specific rules. When differentiating the cotangent function, remember that the derivative of \( \cot(u) \) with respect to \( u \) is \( -\csc^2(u) \). Here \( \csc(u) \) is the cosecant of \( u \), defined as \( \csc(u) = \frac{1}{\sin(u)} \). Understanding these basic rules helps to apply them when working with more complex functions involving these trigonometric ratios. In this case, by differentiating \( y = \cot(u) \), we found that \( \frac{dy}{du} = -\csc^2(u) \). This outcome is pivotal because it sets the stage for utilizing the chain rule, where we will combine this derivative with that of the inner function.

Composite functions are made by merging two functions together. When you have one function plugged into another, this is known as a nested function or a composite function.In the exercise, we have \( y = \cot(\pi - \frac{1}{x}) \), where \( \pi - \frac{1}{x} \) is nested inside the \( \cot \) function. The inner part, \( u = \pi - \frac{1}{x} \), acts as a placeholder within the cotangent function.To effectively find the derivative, you first need to identify the inner and outer functions separately:

• Outer function: \( y = \cot(u) \)
• Inner function: \( u = \pi - \frac{1}{x} \)

By breaking down these components, you make complex derivatives more manageable and prepare for applying the chain rule, blending derivatives of both functions.

Differentiation involves various techniques that help you find the rate of change of a function with respect to its variables. With functions like the one in our exercise, combination techniques come into play.A powerful method for dealing with composite functions is the chain rule. It allows you to differentiate functions layer by layer. When using the chain rule, start by taking the derivative of the outer function, then multiply it by the derivative of the inner function.In our example, we first identified:

• The derivative of the outer function, \( \cot(u) \), as \( -\csc^2(u) \)
• The derivative of the inner function, \( \pi - \frac{1}{x} \), as \( \frac{1}{x^2} \)

Applying the chain rule, we combined these derivatives: \[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = -\csc^2\left(\pi - \frac{1}{x}\right) \times \frac{1}{x^2} \]This strategy simplifies the differentiation process, making it easier to obtain the final derivative of composite functions in terms of the original variable.