Worked example: Implicit differentiation | Advanced derivatives | AP Calculus AB | Khan Academy

TL;DR

Learn how to find the derivative of a function using implicit differentiation.

Transcript

Let's get some more practice doing implicit differentiation. So let's find the derivative of y with respect to x. We're going to assume that y is a function of x. So let's apply our derivative operator to both sides of this equation. So let's apply our derivative operator. And so first, on the left hand side, we essentially are just going to apply ... Read More

Key Insights

• 😀 Implicit differentiation is used when it is not possible to solve for y explicitly in terms of x.
• 📏 The chain rule is utilized in implicit differentiation to find the derivative of a function.
• 🆘 The process of distributing and simplifying the equation helps solve for the derivative of y with respect to x.

Questions & Answers

Q: What is implicit differentiation?

Implicit differentiation is a technique used to find the derivative of a function when it is not possible to solve for y explicitly in terms of x.

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

In implicit differentiation, the chain rule is used to find the derivative of x minus y squared with respect to x. This involves multiplying the derivative of the outer function (x minus y squared) by the derivative of the inner function (x minus y) with respect to x.

Q: How is the derivative of y with respect to x solved for?

To solve for the derivative of y with respect to x, the equation is simplified by distributing 2x minus 2y onto each term. Then, 2x minus 2y plus dy dx is subtracted from both sides, and the equation is further simplified to solve for dy dx.

Q: What is the final expression for the derivative of y with respect to x?

The final expression for the derivative of y with respect to x is 2y minus 2x plus 1 over 2y minus 2x minus 1.

Summary & Key Takeaways

• This content teaches implicit differentiation and how to find the derivative of a function using this method.

• The chain rule is used to find the derivative of x minus y squared with respect to x.

• The derivative of y with respect to x is solved for by simplifying the equation and dividing both sides.