How to Find Partial Derivatives Using Implicit Differentiation

TL;DR

To find partial derivatives of a function implicitly, apply the implicit function theorem by expressing the function equation in the form f(x, y, z) = 0 and then differentiate with respect to x and y. This process includes isolating terms and simplifying results to derive the partial derivatives of z concerning x and y, offering systematic methods for analyzing functions defined implicitly.

Transcript

so here we have an equation in terms of x y and z and we're asked to find the partial derivative of z with respect to x and the partial derivative of z with respect to y now we're going to use the implicit function theorem to get the answer and then we're going to talk about another way that we can get the answer so let's go ahead and begin the fir... Read More

Key Insights

• ❓ The implicit function theorem provides a systematic approach to finding partial derivatives when a function is defined implicitly.
• 🙃 Implicit differentiation is an alternative method to find the same partial derivatives by differentiating both sides of the equation with respect to the desired variable.
• 🫡 Partial derivatives can be used to analyze the sensitivity of a function with respect to changes in its variables.
• 🫡 The process shown in the video can be applied to various examples by identifying the equation and differentiating it with respect to the desired variables.
• ❓ Understanding the implicit function theorem and implicit differentiation is essential for solving more complex mathematical problems.
• 🍉 The process of finding partial derivatives involves differentiating each term in the equation while treating other variables as constants.
• 😑 Canceling out common terms and simplifying the resulting expressions can help make the partial derivatives more manageable.

Summary & Key Takeaways

• The video discusses the process of finding the partial derivatives of a function using the implicit function theorem.

• It demonstrates two different methods to find the partial derivatives of z with respect to x and y, by first using the implicit function theorem and then using implicit differentiation.

• The video also provides an additional example to further illustrate the concept.