91Ó°ÊÓ

The Quotient Rule is an essential tool when finding the derivative of a function that is expressed as a ratio of two separate functions. It allows us to differentiate these functions efficiently. Suppose we have a function \( y = \frac{u}{v} \), where both \( u \) and \( v \) are differentiable functions of \( x \). The quotient rule formula is:

• \( \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \)

This formula helps us find the derivative of the given function by calculating the derivative of the numerator \( u \) and the denominator \( v \), then plugging these into the formula.
In the given exercise, we use the quotient rule to find the derivative of \( y = \frac{x^6 + 8}{16x^2} \) by letting \( u = x^6 + 8 \) and \( v = 16x^2 \).
The derivatives are: \( \frac{du}{dx} = 6x^5 \) and \( \frac{dv}{dx} = 32x \). Plugging these into the quotient rule gives us a derivative that can then be simplified further.

Integral calculation is pivotal when determining arc lengths. Once the derivative of a function is known, the arc length \( L \) of a curve from point \( a \) to point \( b \) can be found using the integral:

• \( L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \)

This formula accounts for the small segments along a curve, summing them to provide the total length.
In our exercise, the derivative \( \frac{dy}{dx} = \frac{6x - x^3}{16} \) was calculated previously. Plugging this into the arc length formula, we get:

• \( L = \int_{2}^{3} \sqrt{1 + \left( \frac{6x - x^3}{16} \right)^2} \, dx \)

The complexity arises in simplifying and evaluating this integral, especially when the expression under the square root isn’t straightforward. Numerical methods or computational tools might be necessary to solve it practically, resulting in the precise arc length of the curve.

Simplifying derivatives is crucial to making further mathematical operations easier. Complex derivatives, if left unrefined, can complicate calculations such as integral evaluation.
Take the derivative \( \frac{dy}{dx} = \frac{6x^4 - x^6}{16x^3} \). At first glance, it might seem intricate, but interestingly, it can be simplified. We factor out \( x^4 \) from the numerator, yielding:

• \( \frac{x^4(6 - x^2)}{16x^3} = \frac{x(6 - x^2)}{16} \)

This simplification transforms the expression into an easier-to-manage form: \( \frac{dy}{dx} = \frac{6x - x^3}{16} \).
The simplified form avoids potential pitfalls in further calculations and allows for a streamlined approach to integration when finding the arc length.