Final answer:
The partial derivative of the function f(x, y, z) with respect to x is 5x/(√(5x²+6y²+z²)).
Step-by-step explanation:
The student is asking for the partial derivative of the function f(x, y, z)=√(5x²+6y²+z²) with respect to x, which is represented as fₓ(x, y, z). To find this derivative, we apply the chain rule. First, we consider u = 5x²+6y²+z² and f(u) = √u. Then we take the derivative of f with respect to u and multiply by the derivative of u with respect to x. The derivative of f with respect to u is (1/2)u^(-1/2) = 1/(2√u), and the derivative of u with respect to x is 10x. Therefore, the partial derivative fₓ(x, y, z) is (10x)/(2√(5x²+6y²+z²)), which simplifies to 5x/(√(5x²+6y²+z²)).