Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{3 x^{3}+4 x} $$

Step-by-Step Solution

Answer

The derivative of the given function is \( y' = (9x^{2}+4)/(3(3x^{3}+4x)^{2/3}) \).

Step 1: Rewrite the Function

The first step is to rewrite the original function in a form that makes it easier to differentiate. We can rewrite the cubic root as a power of 1/3. So \( y=(3x^{3}+4x)^{1/3} \).

Step 2: Use the General Power Rule

Next, we apply the General Power Rule, which says the derivative of \( x^n \) is \( nx^{n-1} \). Here we have \( (3x^{3}+4x)^{1/3} \), so \( n = 1/3 \) and \( x \) is the entire inside function \( 3x^3 + 4x \). Applying the rule gives us the derivative \( y' = \frac{1}{3}(3x^{3}+4x)^{-2/3} \).

Step 3: Use the Chain Rule

We need to remember that when applying the power rule in Step 2, we treated the entire equation \( (3x^{3}+4x) \) as the base \( x \). Therefore, we have to multiply by the derivative of this inner function, a process known as using the Chain Rule. The derivative of \( 3x^{3}+4x \) is \( 9x^{2}+4 \). Therefore, the derivative of the original function is \( y' = \frac{1}{3}(3x^{3}+4x)^{-2/3} \cdot (9x^{2}+4) \).

Step 4: Simplify the result

Finally, simplify the result to get the final answer which is \( y' = (9x^{2}+4)/(3(3x^{3}+4x)^{2/3}) \).