91Ó°ÊÓ

The Quotient Rule is a handy tool for differentiating functions expressed as fractions. It helps when you have one function divided by another, like \( \frac{u}{v} \). The rule is summarized as: \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \] Where:

• \( u \) is the numerator function.
• \( v \) is the denominator function.
• \( u' \) and \( v' \) are their respective derivatives.

In the problem, we utilize the Quotient Rule while differentiating the expression \( \frac{2-y}{x+3} \) to find \( \frac{d^2 y}{dx^2} \). Here:

• \( u = 2 - y \).
• \( v = x + 3 \).
• The derivative \( u' \) equals \(-\frac{dy}{dx} \), as derived from the chain rule.
• \( v' = 1 \), since the derivative of \( x+3 \) is straightforward.

By applying the Quotient Rule, you can effectively differentiate functions of this form. Remember to substitute back any expressions like \( \frac{dy}{dx} \) with their calculated values to simplify results.

The Product Rule is a method for differentiating products of two functions. If you have a product \( uv \), where \( u \) and \( v \) are both functions of \( x \), the rule states: \[\frac{d}{dx}(uv) = u'v + uv'\] Where:

• \( u \) and \( v \) are the two functions you are differentiating.
• \( u' \) and \( v' \) are their derivatives.

Looking at our original expression \( xy = 2x - 3y \), we employed the Product Rule to differentiate the term \( xy \) explicitly:

• With \( x \) and \( y \) treated as the two functions, \( u' = 1 \) since the derivative of \( x \) is 1.
• \( v \), the derivative of \( y \), is \( \frac{dy}{dx} \).
• Applying the rule, it becomes \( xy' + x\frac{dy}{dx} \).

This approach simplifies the differentiation of products in implicit functions, essential in our exercise where each term factors into the re-arranged equation.

The Chain Rule is crucial when dealing with composite functions. It allows you to find the derivative of functions nested within one another. The formula is as follows: \[\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)\] Where:

• \( f \) is the outer function, applied to \( g(x) \).
• \( g \) is the inner function dependent on \( x \).

In the current exercise, the Chain Rule is used when differentiating the term involving \( y \). Because \( y \) is implicitly related to \( x \), when you differentiate such terms like \( 3y \), you must include the derivative of \( y \), which is \( \frac{dy}{dx} \). The derivative becomes \( 3 \cdot \frac{dy}{dx} \) applying the Chain Rule to account for the dependency of \( y \) on \( x \). This step is key when differentiating expressions in implicit differentiation, ensuring all variables related indirectly to \( x \) are appropriately differentiated.