Find the derivative of y=(4⋅x2+8)1x . Recall that ln(x) is entered as log(x).

To find the derivative of the function y=(4x²+8)^(1/x), we will use both the chain rule and the power rule.

Step 1: Rewrite the function First, we rewrite the function in a form that allows us to apply the chain rule and the power rule. We can rewrite the function as y=e^(ln((4x²+8)^(1/x))).

Step 2: Apply the chain rule The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, our outer function is e^u and our inner function is ln((4x²+8)^(1/x)). The derivative of e^u is e^u, so we get y'=e^(ln((4x²+8)^(1/x))) * d/dx[ln((4x²+8)^(1/x))].

Step 3: Differentiate the inner function To differentiate the inner function, we use the power rule, which states that the derivative of u^n is n*u^(n-1)*u'. Here, our u is (4x²+8) and our n is 1/x. So, we get d/dx[ln((4x²+8)^(1/x))] = (1/x) * (4x²+8)^(1/x - 1) * d/dx[(4x²+8)] - ln((4x²+8))/x².

Step 4: Differentiate (4x²+8) The derivative of (4x²+8) is 8x.

Step 5: Substitute back into the equation Substituting back into the equation gives us y' = e^(ln((4x²+8)^(1/x))) * [(1/x) * (4x²+8)^(1/x - 1) * 8x - ln((4x²+8))/x²].

Step 6: Simplify the equation Finally, we simplify the equation to get the derivative of the function. The final derivative is y' = (4x²+8)^(1/x) * [(8x/(x*(4x²+8)) - ln((4x²+8))/x²].

This is the derivative of the function y=(4x²+8)^(1/x).