Q. 23 - Calculus

Question

Use the definition of the partial derivative to find the partial derivatives specified in Exercises 23–26.

$f_y\left(2,1\right)$ when $f\left(x,y\right)=9-x^2-y^2$.

Step-by-Step Solution

Answer

The partial derivative is $-2$.

Step 1: Given

The given function is: $f\left(x,y\right)=9-x^2-y^2$.

Step 2: To find

We have to find $f_y\left(2,1\right)$.

Step 3: Calculation

Differentiate both sides with respect to $y$ and then plug x = 2 and y = 1.

$$\begin{gathered} f\left(x,y\right)=9-x^2-y^2 \\ {f}_y\left(x,y\right)=\frac{\partial }{\partial y}\left(9-x^2-y^2\right) \\ {f}_y\left(x,y\right)=0-0-2y \\ {f}_y\left(x,y\right)=-2y \\ {f}_y\left(2,1\right)=-2\left(1\right) \\ {f}_y\left(2,1\right)=-2 \end{gathered}$$