Evaluate the derivative of the following functions.
f(y) = tan^-1(2y² - 4)

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Identify the function for which you need to find the derivative: \( f(y) = \tan^{-1}(2y^2 - 4) \). This is an inverse trigonometric function, specifically the arctangent function.

Recall the derivative formula for the arctangent function: \( \frac{d}{dx} [\tan^{-1}(u)] = \frac{1}{1 + u^2} \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \).

In this problem, \( u = 2y^2 - 4 \). First, find the derivative of \( u \) with respect to \( y \): \( \frac{du}{dy} = \frac{d}{dy}(2y^2 - 4) \).

Calculate \( \frac{du}{dy} \): Differentiate \( 2y^2 \) to get \( 4y \), and note that the derivative of a constant \(-4\) is 0. Thus, \( \frac{du}{dy} = 4y \).

Substitute \( u = 2y^2 - 4 \) and \( \frac{du}{dy} = 4y \) into the derivative formula: \( \frac{d}{dy}[\tan^{-1}(2y^2 - 4)] = \frac{1}{1 + (2y^2 - 4)^2} \cdot 4y \).

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Key Concepts

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

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Chain Rule

The chain rule is a fundamental technique for differentiating composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential when dealing with functions that are nested within one another, such as the arctangent function in the given problem.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are the functions that reverse the action of the standard trigonometric functions. For example, if y = tan(x), then x = arctan(y). The derivatives of these functions have specific formulas, which are crucial for finding the derivative of functions like f(y) = tan^(-1)(u), where u is a function of y. Understanding these derivatives is key to solving problems involving inverse trigonometric functions.

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