91Ó°ÊÓ

The Chain Rule is a fundamental concept in calculus used for differentiating composite functions. When you have a function within another function, the Chain Rule helps us find the derivative in a step-by-step manner.
For example, consider the expression \( \tan(x + y) \) from the exercise. Here, \( \tan(u) \) is the outer function, and \( u = x + y \) is the inner function.
The Chain Rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

• Step 1: Differentiate the outer function \( \tan(u) \) to get \( \sec^2(u) \).
• Step 2: Multiply it by the derivative of the inner function \( (1 + \frac{dy}{dx}) \) since we have \( x + y \) where \( y \) is a function of \( x \).

This results in \( \sec^2(x+y) \cdot (1 + \frac{dy}{dx}) \).
The Chain Rule simplifies the differentiation of complex expressions and is essential when dealing with implicit functions.

Derivative Evaluation involves determining the specific value of a derivative at a point. This is a practical step after differentiating a function.
Once you have the formula for the derivative, you substitute the given values to find the exact rate of change.
In this exercise, after finding \( \frac{dy}{dx} = \cos^2(x+y) - 1 \), we evaluate it at the point \((0,0)\).
Plug in \( x = 0 \) and \( y = 0 \) into the derivative expression:

• Calculate \( \cos^2(0 + 0) = \cos^2(0) \).
• Since \( \cos(0) = 1 \), we have \( \cos^2(0) = 1 \).

So, \( \frac{dy}{dx} = 1 - 1 = 0 \) at \((0,0)\).
This step confirms the specific behavior of the function at the given point, showing no change in \( y \) with respect to \( x \) here.

An Implicit Function is one where the relationship between \( x \) and \( y \) is given in a form that isn’t explicitly solved for one variable in terms of the other. Instead of \( y = f(x) \) or \( x = g(y) \), you have a relationship like \( \tan(x + y) = x \).
Implicit differentiation is applied here because the equation isn’t solved for \( y \) directly. Instead, both variables appear together under a function, necessitating a different approach than explicit differentiation.
For the given problem, we still derive \( \frac{dy}{dx} \) by following the rules of differentiation applicable to each term while considering both variables are interdependent:

• Differentiate each part treating \( y \) as a function of \( x \).
• Apply the Chain Rule and product rule where necessary.
• Combine and simplify terms to isolate \( \frac{dy}{dx} \).

This showcases the power of implicit differentiation in handling complex equations where direct solving isn’t feasible.