How to Use Implicit Differentiation for Derivatives

TL;DR

To use implicit differentiation, differentiate both sides of an equation with respect to x, treating y as a function of x. For each term involving y, include dy/dx, isolate dy/dx by moving other terms to the opposite side, and then factor it out to solve for dy/dx.

Transcript

in this video we're going to talk about how to do implicit differentiation so let's say if you're given a problem that looks like this x squared plus y cubed minus 4x is equal to eight and then you're told to find d y over dx so what do you need to do in order to find d y over dx in this problem what you need to do first is you need to differentiat... Read More

Key Insights

• 🙃 Implicit differentiation involves differentiating both sides of an equation with respect to x.
• ☺️ The power rule is used to differentiate terms with x variables.
• 🐞 Terms with y variables have dy/dx added to them.
• 🐞 Factors without dy/dx are moved to the other side of the equation to isolate dy/dx.
• 👻 Implicit differentiation allows for finding derivatives of equations that cannot be easily solved using other methods.
• ❣️ The quotient rule is used when differentiating terms with both x and y variables.
• ✋ Implicit differentiation can be used to find higher-order derivatives.

Questions & Answers

Q: What is implicit differentiation?

Implicit differentiation is a method used to find the derivative of an equation with both x and y variables. It involves differentiating both sides of the equation with respect to x, treating y as a function of x.

Q: How do you differentiate terms with x variables?

To differentiate terms with x variables, the power rule is used. For example, the derivative of x^2 is 2x.

Q: How do you differentiate terms with y variables?

When differentiating terms with y variables, dy/dx is added to the equation. For example, the derivative of y^3 is 3y^2 * dy/dx.

Q: How do you isolate dy/dx after differentiation?

To isolate dy/dx, factors without dy/dx are moved to the other side of the equation. Then, dy/dx is factored out and divided by the remaining terms.

Summary & Key Takeaways

• Implicit differentiation involves differentiating both sides of an equation with respect to x, treating y as a function of x.

• The power rule is used to differentiate terms with x variables, and dy/dx is added to terms with y variables.

• After differentiating, factors without dy/dx are moved to the other side of the equation, and dy/dx is factored out to isolate it.