Final answer:
The fifth derivative of the product of five variables with respect to x is 5! times the product of the first four variables. This is calculated through successive differentiation using the product rule. The correct answer is option A
Step-by-step explanation:
If we are given that y is the product of five variables, y = (x₁)(x₂)(x₃)(x₄)(x₅), and we need to find the fifth derivative of y with respect to x, designated as (dy⁵)/(dx⁵), we shall proceed with successive differentiation.
Assuming all x variables are independent of each other and depend only on x, we apply the product rule for differentiation multiple times. Each time we differentiate, one of the x factors will be selected for differentiation, resulting in that factor being reduced by one. We begin this process:
First Derivative:
(dy/dx) = (x₂)(x₃)(x₄)(x₅) + (x₁)(x₃)(x₄)(x₅) + (x₁)(x₂)(x₄)(x₅) + (x₁)(x₂)(x₃)(x₅) + (x₁)(x₂)(x₃)(x₄)
Second through Fifth Derivatives:
Continue differentiating each term until we reach the fifth derivative. By the fifth derivative, all terms will have been differentiated five times resulting in a constant due to the nature of constant derivatives.
Noticing a pattern after the first differentiation, we differentiate four more times to reach the fifth derivative. We will have differentiated each term exactly once in each subsequent derivative. Since there are five terms and each term ends up being differentiated once after five differentiations, we arrive at a factor of 5! along with the remaining term that never gets differentiated, which is the term that originally contained the variable we just differentiated. This results in the answer:
(dy⁵)/(dx⁵) = 5! * (x₁)(x₂)(x₃)(x₄)