91Ó°ÊÓ

The chain rule is a powerful tool in calculus that helps us differentiate composite functions. Consider a function that is composed of two or more functions, such as the one in our original exercise, where we have to differentiate a function like

• \( y = 6(4-x^2)^{-1} \)

To apply the chain rule:

• Identify the outer function, which is the power of \( -1 \) in this case.
• Identify the inner function, which is \( 4-x^2 \).

The chain rule states that the derivative of this composite function is the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function itself. In simplified form, assuming \( u = 4-x^2 \), the derivative becomes

• \( -u^{-2} \times (-2x) = (4-x^2)^{-2} \times (-2x) \)

This method is essential when dealing with any nested derivatives, ensuring each part is broken down step-by-step.

The power rule is a basic differentiation technique used in calculus to determine the derivative of functions that have a variable raised to a power. In the context of our problem,

• we applied it when dealing with the term \((4-x^2)^{-1}\) within the function \(y = 6(4-x^2)^{-1}\).

To apply the power rule, follow these steps:

• Assume \(u = 4-x^2\).
• Then the function becomes \(6u^{-1}\).
• Differentiate \(u^{-1}\) to get \(-u^{-2}\).

This power rule combined with the chain rule allows the derivative to be calculated efficiently. The power rule states that if a function is in the form of \( u^n \), its derivative will be \( nu^{n-1} \), where \( u \) is a differentiable function of \( x \). This application is more intricate when the exponent \( n \) is negative as in our example.

After calculating the derivative of the function, it’s important to evaluate it at a specific point to understand its behavior there. In our exercise, we were given the function \( \frac{dy}{dx} = 12x(4-x^2)^{-2} \)and instructed to substitute \( x = -1 \). By plugging in \( x = -1 \), we calculated:

• \( \frac{dy}{dx} = 12(-1)(4 - (-1)^2)^{-2} = 12(-1)(3)^{-2} \)
• Further simplification yields \( = 12(-1) \cdot \frac{1}{9} = -\frac{12}{9} = -\frac{4}{3} \).

This step is crucial as it translates the abstract concept of differentiation into a concrete number that gives insight into the rate of change at a specific point on the curve.

Verification of a derivative through alternative means is a good practice to ensure accuracy of the results. For this exercise, we used a calculator’s derivative evaluation feature to cross-check our manual calculations.By entering the function \( f(x) = 12x(4-x^2)^{-2} \) into the calculator and substituting \( x = -1 \),the result was \( -\frac{4}{3} \),exactly as per our manual calculations.Verification allows students to gain confidence in their solutions and ensures that computational or algebraic mistakes have not been made. In practice, always consider using technology as a backup to manual problem-solving techniques, helping strengthen understanding through multiple verification approaches.