91Ó°ÊÓ

The chain rule is a powerful technique used in calculus to find derivatives of composite functions. When differentiating a composition of two or more functions, the chain rule plays a critical role. In simple terms, it's like peeling layers of an onion, where you differentiate the outer function and then multiply by the derivative of the inner function.

Here is how it generally works:

• If you have a composite function like \( f(g(x)) \), and you need to differentiate it with respect to \( x \), the chain rule tells us to differentiate \( f \) with respect to \( g \), and multiply by the derivative of \( g \) with respect to \( x \).
• In mathematical notation, this is expressed as \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).

When applying the chain rule to the original exercise, we handled the term \( r^{1/2} \) by differentiating it with respect to \( r \), and then multiplying by \( \frac{dr}{d\theta} \) (since \( r \) itself depends on \( \theta \)). This step effectively connects the changing rate of \( r \) with respect to \( \theta \), ensuring accurate computation.

Partial derivatives are used when dealing with functions of multiple variables. Instead of calculating a derivative with respect to just one variable, partial derivatives provide information about how the function changes as one specific variable changes, keeping others constant.

For example:

• Consider the function \( f(x, y) = x^2 + y^2 \). The partial derivative with respect to \( x \) is obtained by treating \( y \) as a constant, resulting in \( \frac{\partial f}{\partial x} = 2x \).
• Similarly, the partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = 2y \).

In context with our problem, we have an equation linking \( \theta \) and \( r \), where \( r \) is implicitly a function of \( \theta \). The differentiation activities here involved identifying how changes in \( \theta \) affect \( r \), a concept related to partial differentiation, as it's critical in understanding how each variable contributes to the function’s rate of change.

Computing derivatives is a fundamental aspect of calculus, which involves finding the rate at which one quantity changes with respect to another. This computation can be straightforward or involve more complex rules like the product, quotients, or chain rules.

To compute derivatives in our exercise:

• We first differently evaluated each part of the expression \( \theta^{1/2} + r^{1/2} = 1 \).
• We used simple differentiation rules for \( \theta^{1/2} \) which gave us \( \frac{1}{2}\theta^{-1/2} \).
• For \( r^{1/2} \), we applied the chain rule leading to \( \frac{1}{2}r^{-1/2} \cdot \frac{dr}{d\theta} \).

This differentiation was conducted on the concept of implicit differentiation, which stands when an equation defines a function implicitly. By computing derivatives, we essentially unlock the information about how \( r \) changes as \( \theta \) varies, providing the result \( \frac{dr}{d\theta} = -r^{1/2}\theta^{-1/2} \). This answer illuminates the subtle relationships between variables in dynamic systems.