# If {eq}f(x, y) = \dfrac{x^3 - y}{xy - 1} + y^2 + x {/eq}, find {eq}\dfrac{\partial f}{\partial x} {/eq} and {eq}\dfrac{\partial f}{\partial y} {/eq}.

## Question:

If {eq}f(x, y) = \dfrac{x^3 - y}{xy - 1} + y^2 + x {/eq}, find {eq}\dfrac{\partial f}{\partial x} {/eq} and {eq}\dfrac{\partial f}{\partial y} {/eq}.

## Quotient rule of differentiation:

Assume {eq}u(x) {/eq} and {eq}v(x) {/eq} be the two differentiable functions, therefore, the function {eq}f(x) = \dfrac {u}{v}, \ \ v(x) \neq 0 {/eq} is also a differentiable function,

The derivative of the function {eq}f(x) {/eq} with respect to {eq}x {/eq} is given by,

$$f'(x) = \frac {u'v - v'u}{v^2}$$

Partial derivative: It is the derivative of a given function with respect to a given variable, and the other variables are considered as a constant.

Become a Study.com member to unlock this answer!

We have,

$$f(x, y) = \frac{x^3 - y}{xy - 1} + y^2 + x$$

(i) {eq}\dfrac {\partial f}{\partial x} {/eq}:

Differentiate the function with...