91Ó°ÊÓ

When you have a function nested within another function, you need to use the chain rule to differentiate. For example, if you have a function in the form of \( g(f(x)) \), the chain rule allows you to differentiate it by multiplying the derivative of the outer function by the derivative of the inner function. This is expressed mathematically as: \[ (g(f(x)))' = g'(f(x)) \cdot f'(x) \].
In the original exercise, when differentiating \(y^{2/3} \) with respect to \ x \, you need to use the chain rule because \( y \) is a function of \ x \.
So, the differentiation of \(y^{2/3} \) with respect to \ x \ involves multiplying the derivative of \ y^{2/3} \ with respect to \ y \ by \ dy/dx \.
This yields: \[ \frac{d}{dx} ( y^{2 / 3} ) = \frac{2}{3} y^{-1/3} \frac{dy}{dx} \]
Thus, the chain rule is key to successfully differentiating complex functions composed of inner and outer functions.

A derivative represents the rate at which a function is changing at any given point. It's like finding the slope of a function at a particular spot.
For instance, the derivative of \( x^{3/2} \) is found by applying the power rule. The power rule states that if you have \( x^n \), the derivative is \( nx^{n-1} \).
In the exercise, differentiating \( x^{3/2} \) results in: \[ \frac{3}{2} x^{1/2} \] while differentiating a constant yields 0 (since constants don't change).
Remember, the derivative helps in understanding how variables relate and change with respect to each other.

In calculus, \(dy/dx\) is the notation used for the derivative of \ y \ with respect to \ x \. It essentially represents the rate at which \ y \ changes as \ x \ changes.
In the context of our exercise, isolating \( dy/dx \) means solving for \ dy/dx \ in terms of \ x \ and \ y \.
After using the chain rule and moving terms around, the expression for \ dy/dx \ is simplified to showcase how changes in \ x \ affect \ y \ in this specific function.
The final form: \[ \frac{dy}{dx} = - \frac{9}{4} x^{1/2} y^{1/3} \] indicates that both \ x \ and \ y \ contribute to the rate of change of \ y \ with respect to \ x \.