Find the partial derivative of $z=3 x^2 y^3$.
Answer (2)
Find the partial derivative with respect to x by treating y as a constant: ∂ x ∂ z = 6 x y 3 .
Find the partial derivative with respect to y by treating x as a constant: ∂ y ∂ z = 9 x 2 y 2 .
The partial derivatives are ∂ x ∂ z = 6 x y 3 and ∂ y ∂ z = 9 x 2 y 2 .
Explanation
Problem Analysis We are given the function z = 3 x 2 y 3 and we need to find its partial derivatives with respect to x and y . This means we will find how z changes as x changes, keeping y constant, and vice versa.
Partial Derivative with Respect to x To find the partial derivative of z with respect to x , denoted as ∂ x ∂ z , we treat y as a constant and differentiate z with respect to x . Applying the power rule, we have:
Calculating \frac{\partial z}{\partial x} ∂ x ∂ z = ∂ 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, to find the partial derivative of z with respect to y , denoted as ∂ y ∂ z , we treat x as a constant and differentiate z with respect to y . Applying the power rule again, we get:
Calculating \frac{\partial z}{\partial y} ∂ y ∂ z = ∂ 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 Therefore, the partial derivatives are: ∂ x ∂ z = 6 x y 3 ∂ y ∂ z = 9 x 2 y 2
Examples
Partial derivatives are used in many fields, such as physics, engineering, and economics. For example, in thermodynamics, partial derivatives are used to describe how the internal energy of a system changes with respect to temperature and volume. In economics, they are used to analyze how a firm's production changes with respect to labor and capital. Understanding partial derivatives allows us to analyze complex systems by examining the rate of change of one variable while holding others constant, providing valuable insights into the behavior of these systems.
The partial derivatives of the function z = 3 x 2 y 3 are ∂ x ∂ z = 6 x y 3 when differentiating with respect to x , and ∂ y ∂ z = 9 x 2 y 2 when differentiating with respect to y .
;
