Solved: "Find the indicated partial derivative. f(x,y,z)=yx+y+z;fy(2,1,−1)"

College
Calculus and Analysis
Differential Calculus
Derivatives

Felix Fitzgerald

Open question

2022-08-18

Find the indicated partial derivative.

f (x, y, z) = \frac{y}{x} + y + z; f_{y} (2, 1, - 1)

Answer & Explanation

Kaleb Burns

Beginner2022-08-19Added 7 answers

We have
$f (x, y, z) = \frac{y}{x + y + z}$
Use the Quotient Rule and make the following assumptions when differentiating with respect to y: ${(\frac{u}{v})}^{'} = \frac{u^{'} v - u v^{'}}{v^{2}}$ to differentiate the function as a whole.
$f_{y} = \frac{1 (x + y + z) - y (1)}{{(x + y + z)}^{2}} = \frac{x + z}{{(x + y + z)}^{2}} \Rightarrow f_{y} (2, 1, - 1) = \frac{2 - 1}{{(2 + 1 - 1)}^{2}} = \frac{1}{4}$
Final Answer:
$f_{y} (2, 1, - 1) = \frac{1}{4}$

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