Final answer:
The partial derivative ∂z/∂x of the function z = x/(x^5 - y^5) with respect to x is calculated using the quotient rule of differentiation. After simplifying, the correct partial derivative is (4x^5 + y^5)/(x^5 - y^5)^2.
Step-by-step explanation:
The student's question involves computing the partial derivative of a function with respect to x. The function is given as z = x/(x5 - y5). To find the partial derivative of z with respect to x, ∂z/∂x, we need to apply the quotient rule of differentiation, which is: ∂z/∂x = (v(u' - u v'))/v2, where u = x and v = x5 - y5. The derivative u' = dx/dx = 1, and the derivative v' = d(x5 - y5)/dx = 5x4 since y is treated as a constant when taking the partial derivative with respect to x.
Plugging these derivatives into the quotient rule formula:
∂z/∂x = ((x5 - y5)(1) - x(5x4))/(x5 - y5)2
∂z/∂x simplifies to:
∂z/∂x = (x5 - y5 - 5x5)/(x5 - y5)2
∂z/∂x further simplifies to:
∂z/∂x = (- 4x5 + y5)/(x5 - y5)2
Therefore, the correct answer is choice (c): (4x5 + y5)/(x5 - y5)2.