Check the solution to "Find ▽ at the given point. ƒƒ(x,y,z)=x2+y2−2z2+zln⁡x,(1,1,1)"

College
Calculus and Analysis
Differential Calculus
Derivatives

Elleanor Mckenzie

Answered question

2021-03-12

Find $▽$ at the given point. $ƒ (x, y, z) = x^{2} + y^{2} - 2 z^{2} + z \ln x, (1, 1, 1)$

Answer & Explanation

Bentley Leach

Skilled2021-03-14Added 109 answers

Step 1
The gradient of a vector f is the collection of all it's partial derivatives. It is denoted as $▽$ f.
Step 2
Given
$ƒ (x, y, z) = x^{2} + y^{2} - 2 z^{2} + z \ln x$ ,
point (1,1,1)
First of all we need to find partial derivatives,
$f_{x} = 2 x + z * \frac{1}{x}$
$f_{y} = 2 y$
$f_{z} = - 4 z + \ln x$
finally, plugging the point (1,1,1)
$f_{x} = 2 + 1 = 3$
$f_{y} = 2$
$f_{z} = - 4 + \ln (1) = - 4$
Step 3
Hence
$▽ f =< 3, 2, - 4 > .$

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