91Ó°ÊÓ

In calculus, the chain rule is a crucial technique used when differentiating composite functions. This rule allows us to differentiate a function by breaking it into simpler components. If you have a function defined as a composition of two or more functions, the chain rule simplifies the differentiation process.
For instance, consider a function like \[ y^2 \], where you treat \( y \) as a function of \( x \). To differentiate \( y^2 \) with respect to \( x \), you apply the chain rule:

• Take the derivative of the outer function: \[ 2y \]
• Multiply by the derivative of the inner function \( y \):\( \frac{dy}{dx} \)
  

Thus, the derivative of \( y^2 \) with respect to \( x \) is \( 2y \frac{dy}{dx} \). This rule is especially handy in implicit differentiation, where variables are interconnected, like in the equation \( y^2 - ye^x = 12 \), allowing you to handle complex differentiations more efficiently.

The product rule is essential when differentiating products of two functions. In situations where two different functions are being multiplied together, like \( y e^x \), the product rule simplifies the process.
The rule states:- If you have \( u(x) \, v(x) \) as two differentiable functions, then the derivative of their product is:\[ (uv)' = u'v + uv' \]
Applying this to \( y e^x \):

• First function: \( y \)
  
• Second function: \( e^x \)
  
• Derivative is: \( \frac{dy}{dx} \cdot e^x + y \cdot e^x \)
  

Notice how each function is differentiated independently, and then the results are systematically combined. This useful technique allows you to tackle derivatives of more complex functions by utilizing basic differentiation principles.

In differentiation, understanding how constants are treated simplifies many processes. A constant is any number in an equation that does not change as variables shift. When differentiating, constants have a specific rule:
**Rule**: The derivative of any constant is zero.

• Example: If you differentiate \( 12 \), you get zero.

This principle is applied in the given problem, where you differentiate the equation \( y^2 - ye^x = 12 \). The \( 12 \) on the right side, being a constant, differentiates to zero.
This simplifies the differentiation process, enabling you to focus on the more variable terms in the equation. Understanding this concept ensures you can easily spot and handle constant terms efficiently in calculus.