Find Partial
$=3 x^2 y^3$

Find the partial derivative with respect to x : ∂ x ∂ ​ ( 3 x 2 y 3 ) = 6 x y 3 .
Find the partial derivative with respect to y : ∂ y ∂ ​ ( 3 x 2 y 3 ) = 9 x 2 y 2 .
The partial derivative with respect to x is 6 x y 3 ​ .
The partial derivative with respect to y is 9 x 2 y 2 ​ .

Explanation

Problem Analysis We are asked to find the partial derivative of the expression 3 x 2 y 3 . Since the problem does not specify with respect to which variable we should differentiate, we will find both partial derivatives: with respect to x and with respect to y .

Partial Derivative with Respect to x First, let's find the partial derivative with respect to x . We treat y as a constant. Using the power rule, we have: ∂ x ∂ ​ ( 3 x 2 y 3 ) = 3 y 3 ∂ x ∂ ​ ( x 2 ) = 3 y 3 ( 2 x ) = 6 x y 3

Partial Derivative with Respect to y Now, let's find the partial derivative with respect to y . We treat x as a constant. Using the power rule, we have: ∂ y ∂ ​ ( 3 x 2 y 3 ) = 3 x 2 ∂ y ∂ ​ ( y 3 ) = 3 x 2 ( 3 y 2 ) = 9 x 2 y 2

Final Answer Thus, the partial derivative with respect to x is 6 x y 3 and the partial derivative with respect to y is 9 x 2 y 2 .


Examples
Partial derivatives are used in economics to analyze marginal cost and marginal revenue. For example, if a company's cost function is given by C ( x , y ) = 3 x 2 y 3 , where x is the amount of labor and y is the amount of capital, then the partial derivative ∂ x ∂ C ​ = 6 x y 3 represents the marginal cost of labor, and ∂ y ∂ C ​ = 9 x 2 y 2 represents the marginal cost of capital. These partial derivatives help the company make decisions about how much labor and capital to use to minimize costs.

Answered by GinnyAnswer | 2025-07-05

The partial derivative of f ( x , y ) = 3 x 2 y 3 with respect to x is 6 x y 3 and with respect to y is 9 x 2 y 2 .
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Answered by Anonymous | 2025-07-06