91Ó°ÊÓ

The chain rule is a fundamental technique in calculus for finding derivatives of compositions of functions. When dealing with implicit differentiation, the chain rule is particularly useful because you're often treating one variable as a function of another.

In the exercise, when differentiating a term like \( y^2 \), we assume \( y \) is a function of \( x \), say \( y = f(x) \). Thus, the derivative \( d(y^2)/dx \) becomes \( 2y \cdot y' \), where \( y' = dy/dx \). This application of the chain rule helps account for the rate of change of \( y \) as \( x \) changes.

The chain rule can be summarized as:\( \frac{d}{dx}[g(f(x))] = g'(f(x)) \cdot f'(x) \). Whenever you're differentiating a nested function, think "chain rule." It's a tool that unlocks complex derivatives.

Logarithmic differentiation involves taking the derivative of a logarithmic function. In this exercise, we dealt with \( \ln(x/y) \). Logarithm properties and implicit differentiation make it a strategic choice for differentiation, especially when tackling variables in denominators.

For \( \ln(x) \), the derivative is \( 1/x \). When differentiating \( \ln(x/y) \), we apply both the chain rule and this fundamental property, leading to \( (y - x\cdot y')/(xy) \).

• This involves using the quotient rule: \( \ln(a/b) = \ln a - \ln b \).
• The chain rule is used since both \( x \) and \( y \) are functions of \( x \).

Logarithmic differentiation is incredibly powerful for simplifying otherwise difficult derivatives, especially where division or multiplication of functions is involved.

The quotient rule is a method for finding the derivative of a division of two functions, and it's invaluable when dealing with expressions like \( x/y \) in this problem.

The quotient rule states that if you have a function \( h(x) = \frac{u(x)}{v(x)} \), its derivative \( h'(x) \) is given by:\[\left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2}\]
In our problem, \( u = x \) and \( v = y \), leading to the derivative \( (y - x\cdot y')/(xy) \). This captures how each variable affects the division rate.

• The rule effectively manages division in differentiation.
• An alternative is the product rule for products of functions, but here, division directs us to the quotient rule: \((uv)' = u'v + uv'\).

Each rule is a valuable asset in your differentiation toolkit.

The derivative represents the rate of change of a function. In this exercise, you're asked to implicitly differentiate an equation to find \( y' \), the derivative of \( y \) with respect to \( x \).

Understanding how to derive a function is crucial for applying calculus in real-world contexts. The task steps you through differentiating each term individually:

• For \( y^2 \), use the chain rule, yielding \( 2y \cdot y' \).
• For \( \ln(x/y) \), logarithm differentiation combined with the quotient rule gives \((y - x\cdot y')/(xy)\).
• \(-4x \) simplifies directly to \(-4\).

Finally, compile these derivatives to formulate and rearrange an equation to isolate \( y' \). This process underscores the relationship between original variables and their rates of change, a central theme in calculus learning.