Calculating the Directional Derivative of ( e^{x-yz} ) at Point ( A(1,1,-1) ) Directed Towards Point ( B(-356) )

The directional derivative is a fundamental concept in multivariable calculus, representing the rate of change of a function in a specific direction. In this article, we will calculate the directional derivative of the function ( f(x, y, z) e^{x - yz} ) at point ( A(1, 1, -1) ) in the direction of point ( B(-356) ).

Understanding the Given Function and Points

We are given the function f(x, y, z) ( e^{x-yz} ) and two points:

Point ( A ): (1, 1, -1)Point ( B ): (-356, presumably one-dimensional as -356 is a scalar)

Note that point B seems to be specified in a one-dimensional form rather than a three-dimensional space which applies to the function ( f(x, y, z) ). This might be an error or an omission, as the function is a three-dimensional function.

Calculating the Gradient at Point A

The gradient of a function gives us the direction of the maximum rate of change of the function. For ( f(x, y, z) e^{x-yz} ), the gradient is denoted as ( abla f leftlangle frac{partial f}{partial x}, frac{partial f}{partial y}, frac{partial f}{partial z} rightrangle ):

Partial Derivatives

Partial Derivative with Respect to ( x ):

(frac{partial f}{partial x} frac{partial}{partial x} e^{x - yz} e^{x - yz} )

Partial Derivative with Respect to ( y ):

(frac{partial f}{partial y} frac{partial}{partial y} e^{x - yz} -ze^{x - yz} )

Partial Derivative with Respect to ( z ):

(frac{partial f}{partial z} frac{partial}{partial z} e^{x - yz} -ye^{x - yz} )

Gradient Vector at Point A(1, 1, -1)

Evaluating the partial derivatives at point A(1, 1, -1):

( abla f(1, 1, -1) leftlangle e^{1 - 1 - 1} cdot 1, e^{1 - 1 - 1} cdot (-1), e^{1 - 1 - 1} cdot (-1) rightrangle leftlangle e^{-1}, -e^{-1}, -e^{-1} rightrangle )

( abla f(1, 1, -1) leftlangle frac{1}{e}, -frac{1}{e}, -frac{1}{e} rightrangle )

Calculating the Direction Vector (vec{PQ})

To find the directional derivative in the direction of point B, we need the vector (vec{PQ}), where P is point A and Q is point B. Given the coordinates of point A and point B:

Point A: (1, 1, -1)

Point B: (-356, presumably in a one-dimensional context)

In a three-dimensional space, point B should be specified in three coordinates. Let's assume a mistake and proceed with the given point B as (-356, 0, 0).

Vector (vec{PQ} leftlangle -356 - 1, 0 - 1, 0 1 rightrangle leftlangle -357, -1, 1 rightrangle )

However, the problem seems to involve a one-dimensional direction, in which case the coordinates of point B should be interpreted differently. If B is interpreted as a point in three dimensions where the other coordinates are zero (as assumed), we proceed with the given vector (vec{PQ} leftlangle -357 rightrangle ).

Unit Vector (vec{u})

The unit vector (vec{u}) in the direction of (vec{PQ}) is given by:

(vec{u} frac{vec{PQ}}{|vec{PQ}|} leftlangle frac{-357}{357}, frac{0}{357}, frac{0}{357} rightrangle leftlangle -1, 0, 0 rightrangle )

Calculating the Directional Derivative

The directional derivative of ( f ) at point A directed towards the direction vector (vec{PQ}) is given by:

( abla f cdot vec{PQ} leftlangle frac{1}{e}, -frac{1}{e}, -frac{1}{e} rightrangle cdot leftlangle -357 rightrangle )

( abla f cdot vec{PQ} frac{1}{e} cdot (-357) -frac{1}{e} cdot (-357) -frac{1}{e} cdot 0 -357 cdot frac{1}{e} 357 cdot frac{1}{e} 0 0 )

The directional derivative in the direction of (vec{PQ}) is zero, indicating that the direction vector does not align with the gradient vector at point A.

Conclusion

In conclusion, the directional derivative of ( e^{x - yz} ) at point A(1, 1, -1) directed towards the specified direction is zero. This result indicates that the change in the function in the given direction at point A is zero.