Compute the directional derivatives of the following...

Compute the directional derivatives of the following functions along unit vectors at the indicated points in directions parallel to the given vector: (a) $\left.f(x, y)=x^{y},\left(x_{0}, y_{0}\right)=(e, e), d=5 x+12\right\}$ (b) $f(x, y, z)=e^{x}+y z,\left(x_{0}, y_{0}, z_{0}\right)=(1,1,1)$ $\mathrm{d}=(1,-1,1)$ (c) $f(x, y, z)=x y z,\left(x_{0}, y_{0}, z_{0}\right)=(1,0,1)$ $d=(1,0,-1)$

Compute the directional derivatives of the following functions along unit vectors at the indicated points in directions parallel to the given vector:
(a) $\left.f(x, y)=x^{y},\left(x_{0}, y_{0}\right)=(e, e), d=5 x+12\right\}$
(b) $f(x, y, z)=e^{x}+y z,\left(x_{0}, y_{0}, z_{0}\right)=(1,1,1)$ $\mathrm{d}=(1,-1,1)$
(c) $f(x, y, z)=x y z,\left(x_{0}, y_{0}, z_{0}\right)=(1,0,1)$ $d=(1,0,-1)$

Key Concepts

Directional Derivative

In multivariable calculus, the directional derivative represents the rate at which a function changes at a point when moving in a specified direction. It is calculated as the dot product of the function’s gradient and a unit vector indicating the direction, effectively projecting the gradient onto that direction.

Gradient

The gradient of a function is a vector that contains all of its partial derivatives. It points in the direction of the steepest increase of the function, and its magnitude represents the rate of maximum increase. This concept is central to finding directional derivatives since it encapsulates how the function changes along each coordinate axis.

Unit Vector

A unit vector has a length of one and is used exclusively to indicate direction without scaling. In the context of directional derivatives, normalizing the given direction vector into a unit vector ensures that the rate of change is measured per unit distance, keeping the derivative independent of the length of the direction vector.

Dot Product

The dot product is an algebraic operation that multiplies corresponding components of two vectors and sums the results. In computing directional derivatives, the dot product of the gradient and the unit vector extracts the component of the gradient in the direction of interest, yielding the rate of change along that direction.

Partial Derivatives

Partial derivatives measure how a function changes as one of its variables changes while keeping the other variables constant. They form the components of the gradient vector and are essential in analyzing and understanding the local behavior of multivariable functions.

Transcript

00:01 In this problem, we're given a few different functions, and we're given a position to evaluate that function at and a direction.

00:06 And then we want to evaluate the directional derivative in that direction that we're given.

00:11 So in the first part here, f of x, y, is equal to x to the y, and the point that we're evaluating it at is x, y are both equal to e.

00:23 And the direction, and i'm going to write this in terms of a unit vector n, is equal to 1 over 13.

00:31 Times 5 in the x direction and 12 in the y direction.

00:36 This is written in the book as 5i vector plus 12 j vector, and this is the same thing.

00:44 I've just normalized the vector so that when we dot it with the gradient, we don't need to worry about the normalization at all.

00:50 So let's jump into calculating the gradient of f in the x direction.

00:55 This is just going to be equal to y times x times y minus 1, going in the direction of the x unit variable.

01:03 And the y part is a little bit harder.

01:06 I will just note that you can also write this as e to the power log xy, which is also equal to e to the y times log x.

01:21 And that makes it a little bit easier to take the derivative with respect to y.

01:25 So we end up with a factor of log x from the chain rule.

01:30 Times e to the y, log x in the y direction.

01:38 And now if we plug in x equals e and y equal e, this is just equal to e to the e times in the x and y direction, actually.

01:49 The magnitude of each of these is the same.

01:53 And so now if we dot this in with the direction n, so df, d, n, being equal to the direction n, n dotted in with the gradient of f.

02:04 This is just going to give us 17 divided by 13 times e to the e...

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