Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 8
Compute the gradient \(\nabla f\). $$ f(x, y, z)=x^{2} e^{y z} $$
Short Answer
Expert verified
\(abla f = (2x e^{yz}, x^2 z e^{yz}, x^2 y e^{yz})\)
Step by step solution
01
Understand the Gradient
The gradient of a function is a vector composed of its partial derivatives with respect to each variable. For a function of three variables, such as f(x, y, z), the gradient is given by \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\).
02
Compute the Partial Derivative with Respect to x
Find \(\frac{\partial f}{\partial x}\). Since \(f(x, y, z) = x^2 e^{yz}\), when differentiating with respect to \(x\), treat \(y\) and \(z\) as constants:\(\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^2 e^{yz}) = 2x e^{yz}\).
03
Compute the Partial Derivative with Respect to y
Find \(\frac{\partial f}{\partial y}\). When differentiating with respect to \(y\), treat \(x\) and \(z\) as constants:\(\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^2 e^{yz})\). Using the chain rule, this becomes \(x^2 \cdot \frac{\partial}{\partial y} (e^{yz}) = x^2 \cdot z e^{yz}\).
04
Compute the Partial Derivative with Respect to z
Find \(\frac{\partial f}{\partial z}\). When differentiating with respect to \(z\), treat \(x\) and \(y\) as constants:\(\frac{\partial f}{\partial z} = \frac{\partial}{\partial z} (x^2 e^{yz})\). Using the chain rule, this becomes \(x^2 \cdot \frac{\partial}{\partial z} (e^{yz}) = x^2 \cdot y e^{yz}\).
05
Combine the Partial Derivatives
Combine the results of the partial derivatives to form the gradient vector:\(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = \left( 2x e^{yz}, x^2 z e^{yz}, x^2 y e^{yz} \right)\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
partial derivatives
Partial derivatives are essential in multivariable calculus. They show how a function changes as we vary one variable while keeping the others constant.
Specifically, for a function of several variables, like our function \(f(x, y, z) = x^2 e^{yz}\), the partial derivatives with respect to each variable tell us how \(f\) changes when that variable changes, with all other variables held fixed.
For example:
Specifically, for a function of several variables, like our function \(f(x, y, z) = x^2 e^{yz}\), the partial derivatives with respect to each variable tell us how \(f\) changes when that variable changes, with all other variables held fixed.
For example:
- The partial derivative of \(f\) with respect to \(x\) (\frac{\frac{\forall f}{\forall x}}) gives the rate at which \(f\) changes as \(x\) changes.
- The partial derivative with respect to \(y\) tells us how \(f\) changes with respect to \(y\).
- Similarly, the partial derivative with respect to \(z\) shows the rate of change of \(f\) with respect to \(z\).
chain rule
The chain rule is a vital tool in calculus for differentiating composite functions.
In simpler terms, the chain rule helps us find the derivative of a function that is composed of other functions.
When applied to partial derivatives, it allows us to differentiate a function involving multiple variables. For instance, in our function \(f(x, y, z) = x^2 e^{yz}\), we use the chain rule to compute the partial derivatives with respect to \(y\) and \(z\).
This step-by-step application of the chain rule is crucial in computing gradients for complex functions.
In simpler terms, the chain rule helps us find the derivative of a function that is composed of other functions.
When applied to partial derivatives, it allows us to differentiate a function involving multiple variables. For instance, in our function \(f(x, y, z) = x^2 e^{yz}\), we use the chain rule to compute the partial derivatives with respect to \(y\) and \(z\).
- When we find the partial derivative of \(f\) with respect to \(y\), we view \(e^{yz}\) as a composite function.
- Using the chain rule, this becomes \(x^2 \frac{\forall}{\forall y} (e^{yz}) = x^2 \times z e^{yz}\).
This step-by-step application of the chain rule is crucial in computing gradients for complex functions.
vector calculus
Vector calculus extends standard calculus to vector fields.
A key concept in vector calculus is the gradient, which is used to find the direction and rate of the fastest increase of a function.
For a scalar function of multiple variables, like \(f(x, y, z)\), its gradient is a vector composed of its partial derivatives:
abla f = \bigg( \frac{\forall f}{\forall x}, \frac{\forall f}{\forall y}, \frac{\forall f}{\forall z} \bigg).
A key concept in vector calculus is the gradient, which is used to find the direction and rate of the fastest increase of a function.
For a scalar function of multiple variables, like \(f(x, y, z)\), its gradient is a vector composed of its partial derivatives:
abla f = \bigg( \frac{\forall f}{\forall x}, \frac{\forall f}{\forall y}, \frac{\forall f}{\forall z} \bigg).
- In our example, the gradient of \(f\) is \( abla f = \big( 2x e^{yz}, x^2 z e^{yz}, x^2 y e^{yz} \big)\).
- This gradient vector points in the direction of steepest ascent of the function.
- It can be used in many applications, from optimization problems to physics and engineering.
