Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. $$ g(x)=4 \sqrt[3]{x}+2 $$

Step-by-Step Solution

Answer

The derivative of the function \(g(x) = 4\sqrt[3]{x} + 2\) is \(g'(x) = 4/3*x^{-2/3}\).

Step 1: Identify the form of the function

Firstly, recognize that the function can be rewritten as a sum of simpler functions. Here, \(g(x) = 4x^{1/3} + 2\). The function \(f(x) = x^n\) is a basic function with a known formula for the derivative.

Step 2: Apply the power rule

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Thus, using the power rule, the derivative of \(4x^{1/3}\) is \(4*1/3*x^{1/3-1}\) which simplifies to \(4/3*x^{-2/3}\).

Step 3: Take the derivative of the constant function

The derivative of a constant function is zero. Thus, the derivative of constant 2 is 0.

Step 4: Combine the results

Since our function is a sum of two simpler functions, its derivative is the sum of the derivatives of the simpler functions. Therefore, the derivative of our original function \(g(x)\) is \(4/3*x^{-2/3} + 0\) or simply, \(4/3*x^{-2/3}\).