# Implicit derivative of (x^2+y^2)^3 = 5x^2y^2 | Summary and Q&A

284.0K views
January 30, 2013
by
Implicit derivative of (x^2+y^2)^3 = 5x^2y^2

## TL;DR

This video explains how to find the derivative of a relationship between x and y using implicit differentiation.

## Key Insights

• ❣️ Implicit differentiation is a method used to find the derivative of a relationship between x and y without solving for y explicitly.
• 📏 The chain rule and product rule are essential tools in implicit differentiation.
• 🫥 Implicit differentiation allows for the calculation of the slope of the tangent line at any given point.

## Transcript

Once again, I have some crazy relationship between x and y. And just to get a sense of what this might look like, if you plot all the x's and y's that satisfy this relationship, you get this nice little clover pattern. And I plotted this off of Wolfram Alpha. But what I'm curious about in this video, as you might imagine from the title, is to figur... Read More

### Q: What is the purpose of implicit differentiation?

Implicit differentiation allows us to find the derivative of a relationship between x and y without explicitly solving for y in terms of x.

### Q: How is the chain rule applied in implicit differentiation?

The chain rule is used to find the derivative of a term raised to a power by multiplying the power with the derivative of the term.

### Q: What is the product rule and how is it used in implicit differentiation?

The product rule is used to find the derivative of the product of two functions. It involves taking the derivative of the first function times the second function, plus the first function times the derivative of the second function.

### Q: How can the slope of the tangent line be determined using the derivative of y with respect to x?

By plugging in the specific values of x and y at a given point into the derived expression, we can calculate the slope of the tangent line at that point.

## Summary & Key Takeaways

• The video demonstrates how to find the rate at which y is changing with respect to x using implicit differentiation.

• The derivative operator is applied to both sides of the equation, using the chain rule and product rule.

• The final step involves solving for the derivative of y with respect to x, allowing the calculation of the slope of the tangent line.