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

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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.

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.

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}.

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}.

Step 5: Simplify the expression for g'(x) by expanding and combining like terms in the numerator, and ensure the denominator is correctly squared.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function in the form f(x) = u(x)/v(x), the derivative is given by f'(x) = (u'v - uv')/v², where u' and v' are the derivatives of u and v, respectively. This rule is essential for differentiating functions like g(x) = (x^(4/3) - 1) / (x^(4/3) + 1).

Simplification of Derivatives

After finding the derivative of a function, simplification is often necessary to express the result in a more manageable form. This may involve factoring, reducing fractions, or combining like terms. Simplifying the derivative can help in analyzing the behavior of the function, such as identifying critical points and determining intervals of increase or decrease.

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