# Using limit definition of multivariable derivative

• jamiemmt
In summary, the conversation involves someone asking for clarification on how to solve for the derivative using the limit definition, specifically when using the projection function p_k. The person is confused about the role of T (the linear transformation) and how to get the value of the limit.
jamiemmt
Hi, everyone-

I have a quick question. When you solve for the derivative (as a linear transformation) using the limit definition of derivative, how does it go?

For example, let p_k be defined as the projection function from Rn to R, projecting onto kth coordinate of the input value.

lim _(h->0)=\frac{f(x+h)-f(x)}{h}=lim_(h->0)\frac{|h_k|}{h}=??

I see that the linear transformation of h is eqal to the numerator in this case (and that p_k is a linear transformation) but I thought that T had to depend upon x, not h...

Thanks!

what is T, how do you get

\frac{f(x+h)-f(x)}{h}=\frac{|h_k|}{h}

and what is your question it is impossible to understand.

f(x+h)=x_k+h_k
f(x)=x_k
so f(x+h)-f(x)=h_k

so I am looking for T= the linear transformation, or the value of this limit.

## 1. What is the limit definition of multivariable derivative?

The limit definition of multivariable derivative is a mathematical formula used to calculate the rate of change of a function with respect to multiple variables simultaneously. It is based on the concept of limits and involves taking the limit of the function as the variables approach a specific point.

## 2. How is the limit definition of multivariable derivative different from the single variable case?

In the single variable case, the derivative is calculated by taking the limit of the function as the variable approaches a specific point. However, in the multivariable case, there are multiple variables involved, so the limit definition involves taking the limit of the function as all the variables approach a specific point simultaneously.

## 3. Why is the limit definition of multivariable derivative important?

The limit definition of multivariable derivative is important because it allows us to calculate the rate of change of a function with respect to multiple variables, which is necessary in many real-world applications. It also serves as the foundation for more advanced concepts in multivariable calculus.

## 4. How do you use the limit definition of multivariable derivative to calculate the derivative of a function?

To use the limit definition, you first need to find the partial derivatives of the function with respect to each variable. Then, plug these partial derivatives into the formula and take the limit as the variables approach the specific point. The resulting value is the derivative of the function at that point.

## 5. Can the limit definition of multivariable derivative be used for all types of functions?

Yes, the limit definition of multivariable derivative can be used for all types of functions, as long as the function is continuous and differentiable. However, the calculation may become more complicated for more complex functions, and other methods such as the chain rule may be more efficient.

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