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The chain rule is a foundational concept in calculus used to differentiate composite functions. A composite function is essentially a function nested inside another function. The chain rule allows you to break these functions into their simpler components to find the derivative.

Imagine you have a function \( y \) that depends on a function \( u \), and \( u \) depends on \( x \). Mathematically, this is expressed as \( y = f(u) \) and \( u = g(x) \).

• First, find the derivative of \( f(u) \) with respect to \( u \).
• Then, find the derivative of \( u \) with respect to \( x \).
• Finally, multiply these derivatives together.

This process makes differentiating complex functions manageable. In the example of \( y = 2(5x^2 - 1)^{-1} \), the outer function is \( u^{-1} \), where \( u = 5x^2 - 1 \). Differentiating the outer function and multiplying by the derivative of \( u \) provides the result.

The power rule is one of the simplest and most commonly used rules in differentiation. It's a method to find the derivative of expressions where a variable is raised to a power. Suppose you have a basic power function \( y = x^n \). The power rule states:

\[ D_x y = nx^{n-1} \]

What this means is you bring the exponent down, multiply it by the base raised to one less than the original exponent.

• If \( y = x^3 \), the derivative is \( D_x y = 3x^2 \).
• If \( y = x^{-1} \), then \( D_x y = -1x^{-2} \).

In the original problem, the power rule is applied to the re-expressed function \( (5x^2 - 1)^{-1} \). By bringing down the power and reducing the original exponent by one, and then multiplying by the derivative of \( u \), you get the derivative.

Rational functions are quotients of two polynomials, such as \( y = \frac{2}{5x^2 - 1} \). Differentiating rational functions usually involves rules such as the quotient rule and chain rule. However, when the numerator is a constant, the chain rule combined with the power rule can be much simpler.

For these functions, first rewrite the function so the variable is in the numerator by expressing it as a negative power. Then apply the chain rule and power rule in steps:

• Identify the function's structure and re-express if necessary.
• Apply the power rule to the new form of the function.
• Apply the chain rule to handle the inner function’s derivative.

The solution was re-expressed as \( 2 \cdot (5x^2 - 1)^{-1} \), turning it into a manageable expression for applying these differentiation rules.