(a) Find the gradient of f . (b) Evaluate the gradient at...

(a) Find the gradient of $ f $. (b) Evaluate the gradient at the point $ P $. (c) Find the rate of change of $ f $ at $ P $ in the direction of the vector $ u $. $ f(x, y, z) = y^2e^{xyz} $, $ P(0, 1, -1) $, $ u = \big \langle \frac{3}{13}, \frac{4}{13}, \frac{12}{13} \big \rangle $

(a) Find the gradient of $ f $.
(b) Evaluate the gradient at the point $ P $.
(c) Find the rate of change of $ f $ at $ P $ in the direction of the vector $ u $.

$ f(x, y, z) = y^2e^{xyz} $, $ P(0, 1, -1) $, $ u = \big \langle \frac{3}{13}, \frac{4}{13}, \frac{12}{13} \big \rangle $

Key Concepts

Gradient

The gradient is a vector that contains all the first-order partial derivatives of a function. It points in the direction of the steepest ascent of the function and its magnitude corresponds to the rate of maximum increase at that point. In multivariable calculus, the gradient is fundamental for understanding how a function changes in space.

Partial Derivatives

Partial derivatives measure how a multivariable function changes with respect to one variable at a time while keeping the other variables constant. They are the building blocks for the gradient and are crucial for performing further analyses, such as finding the directional derivative or applying optimization techniques.

Directional Derivative

The directional derivative gives the rate at which a function changes at a point in a specified direction. It is calculated as the dot product of the gradient and a unit vector pointing in the given direction. This concept extends the idea of a derivative from single-variable calculus to functions of several variables, allowing one to quantify change in any direction.

Chain Rule

The chain rule is a method for differentiating compositions of functions. When a function contains nested functions, such as an exponential function with a product of variables in its exponent, the chain rule allows one to compute the derivative efficiently and correctly. It is essential in multivariable calculus to handle complex function compositions.

Unit Vector

A unit vector is a vector with a magnitude of one, used to indicate direction without altering the scale of the measurement. In the context of directional derivatives, using a unit vector ensures that the computed derivative reflects the true rate of change of the function in that specific direction, independent of the length of the direction vector.

Transcript

00:01 So for this, we want to find the gradient vector.

00:05 And to do this, we want to find the gradient of f.

00:08 So we know that the gradient is the partial derivative of x, partial derivative of x, partial derivative of y, partial derivative of z, and it's the vector.

00:18 So this is going to be the gradient of f, and it will equal, we want to take the partial derivative with respect to x first.

00:28 And that's going to ultimately give us y cubed z e to the x y z then we take the partial derivative with respect to y and get 2 y e to the x y z and then um that's plus x y squared z e to the x y z and lastly we get x y cubed e to the x y z that right there is our gradient of f a vector we could simplify this further by factoring out an e to the xyz when we do that we'll just pull this out right here that will be our final answer for the gradient of f then for part b what we're wanting to do is evaluate the gradient at point p we know that that p is 0, 1, negative 1, trying to x, y, and z respectively.

01:55 So what we'll end up getting for f, the gradient of f of 0, 1, negative 1 is going to be 1 cubed.

02:11 That's just 1 times 1 times e to the 0, which is just 1.

02:19 Oh, and this is going to be a negative one.

02:22 Then right here, we get e to the zero, which is just one times two times one plus zero times a bunch of things.

02:37 So that's just two times one.

02:40 Let's clear this up.

02:41 So this is going to be two, and this will be a negative one right here...