Computing the Partial Derivatives

Oct 13, 2025 by windailys.com 42 views

Introduction

In this article, we will delve into the world of multivariable calculus and explore the concept of the gradient of a function. Specifically, we will be working with the function $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ defined as $f(x, y, z) = y^2 z + (x+y)^2 + (z-1)^2$ . Our goal is to compute the gradient of this function, which will provide us with valuable information about the behavior of the function.

What is the Gradient?

The gradient of a function is a vector-valued function that points in the direction of the greatest rate of increase of the function at a given point. In other words, it is a measure of the rate of change of the function in each of the input variables. The gradient is denoted by the symbol $\nabla$ and is computed as the partial derivative of the function with respect to each of the input variables.

Computing the Gradient

To compute the gradient of the function $f(x, y, z) = y^2 z + (x+y)^2 + (z-1)^2$ , we need to take the partial derivative of the function with respect to each of the input variables $x$ , $y$ , and $z$ . This can be done using the following formulas:

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (y^2 z + (x+y)^2 + (z-1)^2)

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (y^2 z + (x+y)^2 + (z-1)^2)

\frac{\partial f}{\partial z} = \frac{\partial}{\partial z} (y^2 z + (x+y)^2 + (z-1)^2)

Computing the Partial Derivatives

Let's start by computing the partial derivative of the function with respect to $x$ .

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (y^2 z + (x+y)^2 + (z-1)^2)

= 2(x+y)

Next, let's compute the partial derivative of the function with respect to $y$ .

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (y^2 z + (x+y)^2 + (z-1)^2)

= 2y(z+1) + 2(x+y)

Finally, let's compute the partial derivative of the function with respect to $z$ .

\frac{\partial f}{\partial z} = \frac{\partial}{\partial z} (y^2 z + (x+y)^2 + (z-1)^2)

= y^2 + 2(x+y)(z-1)

The Gradient

Now that we have computed the partial derivatives of the function with respect to each of the input variables, we can write the gradient of the function as:

\nabla f = \left( \begin{array}{l} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \end{array} \right)

= \left( \begin{array}{l} 2(x+y) \\ 2y(z+1) + 2(x+y) \\ y^2 + 2(x+y)(z-1) \end{array} \right)

Conclusion

In this article, we have computed the gradient of the function $f(x, y, z) = y^2 z + (x+y)^2 + (z-1)^2$ . The gradient is a vector-valued function that points in the direction of the greatest rate of increase of the function at a given point. We have shown that the gradient can be computed by taking the partial derivative of the function with respect to each of the input variables. The gradient is an important concept in multivariable calculus and has many applications in fields such as physics, engineering, and economics.

References

[1] "Calculus" by Michael Spivak
[2] "Multivariable Calculus" by James Stewart
[3] "Vector Calculus" by Peter Baxandall

Q: What is the gradient of a multivariable function?

A: The gradient of a multivariable function is a vector-valued function that points in the direction of the greatest rate of increase of the function at a given point. It is a measure of the rate of change of the function in each of the input variables.

Q: How is the gradient computed?

A: The gradient is computed by taking the partial derivative of the function with respect to each of the input variables. This involves differentiating the function with respect to each variable and then combining the results into a single vector.

Q: What is the significance of the gradient?

A: The gradient is an important concept in multivariable calculus and has many applications in fields such as physics, engineering, and economics. It is used to optimize functions, find the maximum or minimum of a function, and determine the direction of the greatest rate of increase or decrease.

Q: How is the gradient used in optimization problems?

A: The gradient is used in optimization problems to find the maximum or minimum of a function. By following the direction of the gradient, one can move in the direction of the greatest rate of increase or decrease, which can help to find the optimal solution.

Q: What are some common applications of the gradient?

A: Some common applications of the gradient include:

Optimization: The gradient is used to optimize functions, find the maximum or minimum of a function, and determine the direction of the greatest rate of increase or decrease.
Physics: The gradient is used to describe the behavior of physical systems, such as the flow of fluids and the motion of particles.
Engineering: The gradient is used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: The gradient is used to model economic systems and make predictions about economic behavior.

Q: What are some common mistakes to avoid when working with the gradient?

A: Some common mistakes to avoid when working with the gradient include:

Not checking the domain of the function: Make sure to check the domain of the function before computing the gradient.
Not checking the differentiability of the function: Make sure to check the differentiability of the function before computing the gradient.
Not using the correct formula for the gradient: Make sure to use the correct formula for the gradient, which involves taking the partial derivative of the function with respect to each of the input variables.

Q: What are some common tools and techniques used to work with the gradient?

A: Some common tools and techniques used to work with the gradient include:

Calculus: Calculus is used to compute the gradient and optimize functions.
Linear algebra: Linear algebra is used to represent the gradient as a vector and perform operations on it.
Numerical methods: Numerical methods, such as gradient descent, are used to approximate the gradient and optimize functions.

Q: What are some common challenges when working with the gradient?

A: Some common challenges when working with the gradient include:

Computational complexity: Computing the gradient can be computationally intensive, especially for large and complex functions.
Numerical instability: Numerical methods used to approximate the gradient can be unstable and produce inaccurate results.
Optimization problems: Optimization problems can be challenging to solve, especially when the function is non-convex or has multiple local optima.

Conclusion

In this article, we have answered some frequently asked questions about the gradient of a multivariable function. We have discussed the significance of the gradient, how it is computed, and some common applications and challenges. We hope that this article has provided a helpful overview of the gradient and its uses in multivariable calculus.