Total Differential of z = e^(xy)*sin(x) Multivariable Calculus

TL;DR

The total differential of a function of two variables can be found using the derivative of each variable with respect to X and Y, which is then plugged into a formula.

Transcript

function of two variables that we're going to find what's called the total differential so the total differential is given by the following formula so it ZZ is equal to the derivative of this function with respect to X so del C del X DX plus the derivative of this function with respect to Y dy so tau Z Y so all we have to do to find the total diffe... Read More

Key Insights

• 🫡 The total differential of a function is determined by finding the derivatives of the function with respect to each variable.
• 📏 The product rule is used to find the derivative of a function with multiple variables multiplied together.
• 📏 The chain rule is utilized when finding the derivative of a function with composed variables.

Questions & Answers

Q: How is the total differential of a function of two variables determined?

The total differential is found by taking the derivative of the function with respect to X and multiplying it with del Z del X, then taking the derivative of the function with respect to Y and multiplying it with del Z del Y. These values are then plugged into the formula ZZ = del C del X DX + tau Z Y dy.

Q: What is the product rule used for?

The product rule is used to find the derivative of a function that has two functions multiplied together. It states that the derivative of the first function times the second function is added to the derivative of the second function times the first function.

Q: When is the chain rule used?

The chain rule is used when taking the derivative of a composed function, where one function is applied to another. It involves taking the derivative of the outside function and evaluating it at the inside function, then multiplying it with the derivative of the inside function.

Q: How is the derivative of a function with respect to Y found?

To find the derivative of a function with respect to Y, the derivative of the first function is multiplied with the second function, and the derivative of the second function is multiplied with the first function. The derivative of the inside function is found by taking the derivative of X Y with respect to Y, which is equal to 1.

Summary & Key Takeaways

• The total differential for a function of two variables can be determined by taking the derivatives of the function with respect to X and Y and plugging them into a formula.

• The product rule and chain rule are used to find the derivative of a function with multiple variables.

• The derivative of a function with respect to X is found by multiplying the derivative of the first function with the second function and adding it to the derivative of the first function times the derivative of the second function.

• The derivative of a function with respect to Y is found by multiplying the derivative of the first function with the second function and adding it to the derivative of the second function times the derivative of the first function.