Q. 44 - Calculus

Question

Find all points where the first-order partial derivatives of the functions in Exercises $43-54$ are continuous. Then use Theorems $12.28$ and $12.31$ to determine the sets in which the functions are differentiable.

 $f(x,y)=\frac{x}{y^2}$

Step-by-Step Solution

Answer

Wherever the fraction $y\ne 0$, these components are continuous. They are ongoing on the subset $S_1=\left\{\right(x,y)\mid y\ne 0\}$, and $f$can be differentiated at any point in$S_1$.

Step: 1 Definition of Partial Derivative:

A approximate solution of a function with many parameters has been its derivative with reference to one of the other variables while the others are maintained constant in mathematics   In linear calculus and differential geometry, partial derivatives are utilised.

Step: 2 Partial derivative function:

Having that,

$f(x,y)=\frac{x}{y^2}$

The derivative functions is

$\frac{a}{dx}f(x,y)=\frac{1}{y^2}$

$\frac{d}{dy}f(x,y)=-\frac{2x}{y^3}$

Step: 3 Finding solution:

Wherever the fraction $y\ne 0$, these components are continuous. They are ongoing on the subset $S_1=\left\{\right(x,y)\mid y\ne 0\}$, and $f$ can be differentiated at any point in $S_1$.