Evaluate the derivative of the following functions.
f(u) = csc^-1 (2u + 1)

Verified step by step guidance

Recognize that the function f(u) = csc^{-1}(2u + 1) involves the inverse cosecant function. The derivative of csc^{-1}(x) with respect to x is -1 / (|x| * sqrt(x^2 - 1)).

Apply the chain rule to differentiate f(u) = csc^{-1}(2u + 1). The chain rule states that if you have a composite function f(g(u)), the derivative is f'(g(u)) * g'(u).

Identify the inner function g(u) = 2u + 1. Differentiate g(u) with respect to u to find g'(u). The derivative of 2u + 1 is 2.

Substitute g(u) = 2u + 1 into the derivative formula for csc^{-1}(x). This gives us -1 / (|2u + 1| * sqrt((2u + 1)^2 - 1)).

Multiply the result from the previous step by g'(u), which is 2, to apply the chain rule. The final expression for the derivative is -2 / (|2u + 1| * sqrt((2u + 1)^2 - 1)).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as csc-1 (x), are the inverses of the standard trigonometric functions. They are used to find angles when given a ratio. Understanding their properties and how they relate to their corresponding functions is crucial for differentiation.

Chain Rule

The chain rule is a fundamental differentiation technique used when differentiating composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). This rule is essential for evaluating the derivative of functions like f(u) = csc-1 (2u + 1), where the inner function is 2u + 1.

Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula: if y = f-1(x), then dy/dx = 1/(df/dy). For inverse trigonometric functions, specific derivatives exist, such as the derivative of csc-1(x), which is -1/(|x|√(x²-1)). Knowing these derivatives is vital for solving the given problem.

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