Derivatives Find and simplify the derivative of the following functions.
g(x) = x⁴/³-1 / x⁴/³+1

Verified step by step guidance

1. Step 1: Identify the function g(x) = \frac{x^{4/3} - 1}{x^{4/3} + 1}. This is a rational function, so we will use the quotient rule to find its derivative.
2. Step 2: Recall the quotient rule for derivatives: if you have a function h(x) = \frac{u(x)}{v(x)}, then h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}. Here, u(x) = x^{4/3} - 1 and v(x) = x^{4/3} + 1.
3. Step 3: Differentiate u(x) and v(x) with respect to x. For u(x) = x^{4/3} - 1, the derivative u'(x) = \frac{4}{3}x^{1/3}. For v(x) = x^{4/3} + 1, the derivative v'(x) = \frac{4}{3}x^{1/3}.
4. Step 4: Substitute u(x), v(x), u'(x), and v'(x) into the quotient rule formula: g'(x) = \frac{(\frac{4}{3}x^{1/3})(x^{4/3} + 1) - (x^{4/3} - 1)(\frac{4}{3}x^{1/3})}{(x^{4/3} + 1)^2}.
5. Step 5: Simplify the expression for g'(x) by expanding and combining like terms in the numerator, and ensure the denominator is correctly squared.