Final Answer:
The four second-order partial derivatives of z = sin(3x) are:
∂²z/∂x² = -9sin(3x),
∂²z/∂y² = 0,
∂²z/∂x∂y = 0,
∂²z/∂y∂x = 0.
Step-by-step explanation:
In the given function z = sin(3x), we are dealing with a single-variable function with respect to x. The second-order partial derivative with respect to x can be found by applying the chain rule twice.
Second-order partial derivative with respect to x:
∂z/∂x = 3cos(3x)
Now, take the derivative of this result with respect to x:
∂²z/∂x² = -9sin(3x)
Second-order partial derivative with respect to y:
Since z is not dependent on y, the second-order partial derivative with respect to y is zero.
Mixed partial derivatives:
The mixed partial derivatives, i.e., ∂²z/∂x∂y and ∂²z/∂y∂x, are also zero as z is not dependent on y.
Understanding partial derivatives is essential in calculus as it helps analyze how a function changes concerning its variables and provides insights into the function's behavior at specific points.