# Implicit partial differentiation

• morsel
In summary, when taking the partial derivative of x with respect to z at the point (1, -1, -3), we must use the product and chain rule in order to correctly differentiate the equation xz + y ln x - x^2 + 4 = 0.
morsel

## Homework Statement

Find $\partial x / \partial z$ at the point (1, -1, -3) if the equation $xz + y \ln x - x^2 + 4 = 0$ defines x as a function of the two independent variables y and z and the partial derivative exists.

## The Attempt at a Solution

$x + y/x \partial x / \partial z - 2x \partial x / \partial z = 0$

Did I do the implicit differentiation correctly? I'm unsure about where to put $\partial x / \partial z$ when I differentiate.

Thanks!

morsel said:

## Homework Statement

Find $\partial x / \partial z$ at the point (1, -1, -3) if the equation $xz + y \ln x - x^2 + 4 = 0$ defines x as a function of the two independent variables y and z and the partial derivative exists.

## The Attempt at a Solution

$x + y/x \partial x / \partial z - 2x \partial x / \partial z = 0$

Did I do the implicit differentiation correctly? I'm unsure about where to put $\partial x / \partial z$ when I differentiate.
No, this isn't correct. When you take the partial of y lnx with respect to z you have to use the product rule (and then the chain rule), and as far as I can tell, you didn't use it.

Once you have differentiated both sides of the equation, solve algebraically for $$\frac{\partial x}{\partial z}$$

I think the ln term is correct. But the first term is not. Here you have to use the product and chain rule.

betel said:
I think the ln term is correct. But the first term is not. Here you have to use the product and chain rule.
I didn't check, but I think you're probably right. Having x be the dependent variable and y and z independent probably threw me off.

## 1. What is implicit partial differentiation?

Implicit partial differentiation is a mathematical concept used to find the rate of change of a multivariable function, where some variables are dependent on others. It involves taking the derivative of a function with respect to one variable while treating the other variables as constants.

## 2. How is implicit partial differentiation different from explicit partial differentiation?

The main difference between implicit and explicit partial differentiation is that in implicit differentiation, the dependent variables are not explicitly defined in terms of the independent variables. This means that the function cannot be easily solved for a particular variable, and the differentiation process is more complex.

## 3. Why is implicit partial differentiation important in scientific research?

Implicit partial differentiation is an essential tool in scientific research because many real-world phenomena involve multiple variables that are interdependent. By using implicit differentiation, scientists can analyze and understand complex relationships between these variables and make predictions about how they will change over time.

## 4. What are some common applications of implicit partial differentiation?

Implicit partial differentiation has many practical applications in fields such as physics, engineering, economics, and biology. It is used to study the relationships between variables in systems and to model and solve problems involving rates of change, optimization, and stability.

## 5. What are some common challenges when using implicit partial differentiation?

One of the main challenges of implicit partial differentiation is that it can be more complicated and time-consuming than explicit differentiation, particularly for more complex functions. Additionally, it requires a deep understanding of mathematical concepts and techniques, and errors can be hard to catch and correct.

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