Final answer:
The derivative of the function f(x) = 6x^6 / √x + 2x^2 / √x is f'(x) = 33x^(9/2) + 3x^(1/2).
Step-by-step explanation:
The student has asked to find the derivative of the function f(x) = 6x6 \/ √x + 2x2 \/ √x. To simplify this expression, we should rewrite the function in terms of x to the power of rational numbers. The square root of x is x to the power of 1/2, so the function becomes f(x)=6x12/2x-1/2 + 2x4/2x-1/2.
This simplifies to f(x) = 6x11/2 + 2x3/2. The derivative of a term with the form axn is anxn-1. Thus, the derivative of our function f with respect to x is:
f'(x) = 6 × (11/2)x11/2-1 + 2 × (3/2)x3/2-1.
Simplifying this, we find that f'(x) = 33x9/2 + 3x1/2. Therefore, the derivative of f(x) is 33x9/2 + 3x1/2.