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When you come across a function represented as a ratio of two expressions, like \( y = \frac{x}{x+8} \), the quotient rule is your go-to differentiation tool. The rule is specially designed for dividing one function by another. Its formula is given by

• \( \left(\frac{v}{w}\right)' = \frac{v'w - vw'}{w^2} \)

where \( v \) and \( w \) are functions of \( x \).
To apply this, first identify the numerator \( v = x \) and the denominator \( w = x+8 \). Differentiate both: \( v' = 1 \) and \( w' = 1 \). Plug these into the quotient rule formula to find \( \frac{dy}{dx} \):

• \( \frac{dy}{dx} = \frac{(1)(x+8) - x(1)}{{(x+8)}^2} = \frac{8}{{(x+8)}^2} \)

Breaking it down, keep an eye on numerator derivatives and simplify the expression step-by-step to maintain clarity.

The chain rule is incredibly powerful for tackling composite functions. These are functions where one function nests inside another, a common scenario in calculus problems. The rule states

• \( (f(g(x)))' = f'(g(x))g'(x) \)

In the given problem, \( y \) depends on \( x \), and \( x \) is a function of \( v \). Thus, you need to differentiate \( y(x(v)) \) using the chain rule. First, compute \( \frac{dy}{dx} \) and \( \frac{dx}{dv} \), then multiply them as follows:

• \( \frac{dy}{dv} = \left( \frac{dy}{dx} \right) \left( \frac{dx}{dv} \right) \)

Once you find \( \frac{dy}{dx} = \frac{8}{{(x+8)}^2} \) and \( \frac{dx}{dv} = 2v \), multiply them to produce the composited derivative result \( \frac{dy}{dv} = \frac{16v}{{(v^2+8)}^2} \). Substitute back \( x = v^2 \) at the end to express \( dy/dv \) entirely in terms of \( v \).

The power rule simplifies the process of differentiating functions raised to a power. It states that for any real number \( n \), the derivative of \( u^n \) is

• \( (u^n)' = nu^{n-1} \cdot \frac{du}{dx} \)

In this context, the power rule is used to differentiate \( x = v^2 \). Here, \( n = 2 \) and \( u = v \), so differentiate using the power rule to get

• \( \frac{dx}{dv} = 2v \)

This step is crucial in connecting the dots for the chain rule application, where each derivative plays a role in breaking down the composite function to its fundamental pieces.