Find the Partial Derivative using the Chain Rule || Multivariable Calculus Example

TL;DR

The video explains how to find a partial derivative with respect to P using the chain rule for functions with multiple variables.

Transcript

okay let's do some math so here we're going to find a partial derivative with respect to P so we have f of x y equal to this function here Ln x cosine Y and then X and Y are defined by these equations here so the solution so the formula is pretty simple so Del F Del p so we're trying to find the partial with respect to P so there's two ways to get ... Read More

Key Insights

• 👻 The chain rule allows finding partial derivatives of functions with multiple variables by considering their interconnectedness.
• 🫡 The partial derivative with respect to P can be obtained by going through X or Y, utilizing the chain rule accordingly.
• 😀 The derivatives of Ln X and cosine Y, as well as the differentiation of equations involving P and Q, are necessary steps in the process.

Questions & Answers

Q: What is the purpose of finding the partial derivative with respect to P using the chain rule?

The chain rule helps find the rate of change of a function when one variable depends on another. In this case, we want to determine how the function changes with respect to P while considering the dependencies of X and Y.

Q: What are the two ways to calculate Del F Del P, going through X or through Y?

To calculate Del F Del P, you can either find Del F Del x times Del X Del P or Del F Del Y times Del Y Del P. Both methods will lead to the same partial derivative but account for different paths to reach P.

Q: Can you explain the process of finding Del X Del P?

Del X Del P requires differentiating the equation X = P^3Q^2 with respect to P. As Q is a constant, the derivative of P^3Q^2 is 3P^2Q^2.

Q: How do we simplify the final partial derivative expression?

After evaluating Del F Del x and Del F Del Y, the final expression simplifies to 3cos(P^2Q^3)/P - 2PQ^3sin(P^2Q^3)ln(P^3Q^2), combining all the terms with appropriate substitutions.

Summary & Key Takeaways

• The video demonstrates how to find the partial derivative with respect to P by going through X or Y, considering X and Y depend on P.

• To find the partial derivative through X, we calculate Del F Del x times Del X Del P, and for the partial derivative through Y, it is Del F Del Y times Del Y Del P.

• The final partial derivative is simplified to 3cos(P^2Q^3)/P - 2PQ^3sin(P^2Q^3)ln(P^3Q^2).