Answered: Computing gradients Compute the…

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Functions And Their Graphs

Section: Chapter Questions

Problem 8PS

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Question

Computing gradients Compute the gradient of the following functions and evaluate it at the given point P.

h(x, y) = ln (1 + x² + 2y²); P(2, -3)

Expert Solution

Step 1

Given that:

The equation is $h (x, y) = \ln (1 + x^{2} + 2 y^{2}); P (2, - 3)$ .

Step 2

By using,

For a function of two variables z = f (x, y), the gradient is the two dimensional vector,

grad f = $<f_{x} (x, y), f_{y} (x, y)>$ .

Chain rule : $\frac{d}{d x} f (g (x)) = f^{'} (g (x)) . \frac{d}{d x} g (x)$

Step 3

By using the above formula,

To find h_x ( x, y) :

Differentiate the given equation with respect to x,

$h_{x} (x, y) = \frac{d}{d x} (\ln (1 + x^{2} + 2 y^{2})) = \frac{1}{(1 + x^{2} + 2 y^{2})} \frac{d}{d x} (1 + x^{2} + 2 y^{2}) . . . . . b y u \sin g c h a i n r u l e = \frac{2 x}{(1 + x^{2} + 2 y^{2})} \Rightarrow h_{x} (x, y) = \frac{2 x}{(1 + x^{2} + 2 y^{2})}$

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