Final Answer:
The gradient of the function f(x, y, z) = 4xe^(y/3)sin(5z) is ∇f = (e^(y/3)sin(5z), 4/3xe^(y/3)cos(5z), 20xe^(y/3)cos(5z)).
Step-by-step explanation:
The gradient, denoted as ∇f, of a scalar function f(x, y, z) is a vector containing its partial derivatives with respect to each variable. For the given function f(x, y, z) = 4xe^(y/3)sin(5z), the partial derivatives are calculated as follows:
∂f/∂x = e^(y/3)sin(5z),
∂f/∂y = (4/3)xe^(y/3)cos(5z),
∂f/∂z = 20xe^(y/3)cos(5z).
Therefore, the gradient ∇f is represented as (e^(y/3)sin(5z), 4/3xe^(y/3)cos(5z), 20xe^(y/3)cos(5z)). Each component of ∇f indicates the rate of change of the function concerning the corresponding variable, providing insights into the function's behavior in different directions in 3D space.
In summary, understanding the gradient is fundamental in multivariable calculus as it helps analyze how a function changes with respect to each input variable. The components of the gradient vector serve as directional derivatives, crucial for comprehending the function's behavior and optimizing various real-world applications.