How to Find dy/dx using Implicit Differentiation Given y^2  y*e^x = 12 (Example with Exponentials)  Summary and Q&A
TL;DR
Use implicit differentiation to find dy/dx, the derivative of y with respect to x.
Key Insights
 🍗 Implicit differentiation is useful when trying to find derivatives in equations where y is not explicitly defined in terms of x.
 😀 The chain rule is crucial in implicit differentiation to account for the derivative of y with respect to x.
 ❣️ The product rule is applied in cases where functions involving both y and x need to be differentiated.
 👻 Factoring out dy/dx simplifies the equation and allows for the isolation of the derivative.
 🐞 The final expression for dy/dx involves both y and e^x terms in the numerator and a combination of y and e^x terms in the denominator.
 ❓ Implicit differentiation can be challenging and requires practice to master.
 📏 Understanding how to use the chain rule and product rule is key to successfully perform implicit differentiation.
Questions & Answers
Q: What is implicit differentiation used for?
Implicit differentiation is used to find the derivative of a function where y is a function of x, especially when it is difficult or impossible to solve for y explicitly.
Q: How is implicit differentiation different from regular differentiation?
Implicit differentiation treats y as a function of x, so the chain rule is applied to account for the derivative of y. Regular differentiation assumes y is explicitly defined in terms of x.
Q: Why is it necessary to use the chain rule in implicit differentiation?
The chain rule is necessary because y is treated as a function of x. When differentiating a term with y, the chain rule accounts for the derivative of y with respect to x.
Q: How does the product rule apply in this implicit differentiation problem?
The product rule is used when differentiating the term involving y and e^x. It involves taking the derivative of y and multiplying it by e^x, then adding the product of y and the derivative of e^x.
Summary & Key Takeaways

Implicit differentiation is used to find the derivative of a function where y is a function of x.

The process involves taking the derivative of both sides of the equation and applying the chain rule.

The final result is dy/dx = ye^x / (2y  e^x).