Solve all the second partial derivatives.&nbsp;f(x,y)=ln⁡(ax+by)

Step 1 f(x,y)=ln⁡(ax+by) fx=∂f∂x=1ax+by∂∂x(ax+by) fx=∂f∂x=aax+by fy=∂f∂y=bax+by Step 2 fxx=∂∂xfx=∂∂x(aax+by) fxx=a2(ax+by)2 fyy=∂∂yfy=−b2(ax+by)2 fxy=∂∂yfx=−ab}{(ax+by)2} fyx=∂∂xfy=−b2(ax+by)2

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emancipezN

Answered question

2021-04-19

Solve all the second partial derivatives.
$f (x, y) = \ln (a x + b y)$

Answer & Explanation

Brittany Patton

Skilled2021-04-21Added 100 answers

Step 1
$f (x, y) = \ln (a x + b y)$
$f_{x} = \frac{\partial f}{\partial x} = \frac{1}{a x + b y} \frac{\partial}{\partial x} (a x + b y)$
$f_{x} = \frac{\partial f}{\partial x} = \frac{a}{a x + b y}$
$f_{y} = \frac{\partial f}{\partial y} = \frac{b}{a x + b y}$
Step 2
$f_{x x} = \frac{\partial}{\partial x} f_{x} = \frac{\partial}{\partial x} (\frac{a}{a x + b y})$
$f_{x x} = \frac{a^{2}}{{(a x + b y)}^{2}}$
$f_{y y} = \frac{\partial}{\partial y} f_{y} = - \frac{b^{2}}{{(a x + b y)}^{2}}$
$f_{x y} = \frac{\partial}{\partial y} f_{x} = - \frac{a b}{}} {{(a x + b y)}^{2}}$
$f_{y x} = \frac{\partial}{\partial x} f_{y} = - \frac{b^{2}}{{(a x + b y)}^{2}}$

Jeffrey Jordon

Expert2022-01-31Added 2605 answers

Answer is given below (on video)

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Find all the second partial derivatives. f(x,y)=\ln(ax+by)

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