Differentiate the function y = $\sqrt[3]{x^2}(2x - x^2)$

1. Analyze the question

We are asked to find the derivative ($\frac{dy}{dx}$) of the function $y = \sqrt[3]{x^2}(2x - x^2)$. To solve this, we need to simplify the expression into a polynomial form (using indices) before applying the power rule of differentiation.

2. Explain the concept

• Indices: Recall that $\sqrt[n]{x^m} = x^{m/n}$. Therefore, $\sqrt[3]{x^2} = x^{2/3}$.
• Power Rule: If $y = ax^n$, then $\frac{dy}{dx} = n \cdot ax^{n-1}$.
• Distributive Law: Multiply the term outside the bracket by each term inside the bracket.

3. Show the solution

Step 1: Rewrite the function using indices $y = x^{2/3}(2x - x^2)$

Step 2: Expand the expression Multiply $x^{2/3}$ by each term inside the bracket: $y = (x^{2/3} \cdot 2x^1) - (x^{2/3} \cdot x^2)$

Using the law of indices ($x^a \cdot x^b = x^{a+b}$):

• For the first term: $x^{2/3} \cdot 2x^1 = 2x^{(2/3 + 1)} = 2x^{5/3}$
• For the second term: $x^{2/3} \cdot x^2 = x^{(2/3 + 2)} = x^{(2/3 + 6/3)} = x^{8/3}$

So, $y = 2x^{5/3} - x^{8/3}$

Step 3: Differentiate with respect to $x$ Apply the power rule $\frac{d}{dx}(ax^n) = nax^{n-1}$:

• Derivative of $2x^{5/3}$: $\frac{5}{3} \cdot 2x^{(5/3 - 1)} = \frac{10}{3}x^{2/3}$

• Derivative of $x^{8/3}$: $\frac{8}{3}x^{(8/3 - 1)} = \frac{8}{3}x^{5/3}$

Combining these: $\frac{dy}{dx} = \frac{10x^{2/3}}{3} - \frac{8x^{5/3}}{3}$

4. Identify the answer

Comparing our result to the given options, we see it matches option B.

5. Tip

When you see roots (like $\sqrt{x}$ or $\sqrt[3]{x^2}$) in a differentiation problem, always convert them to fractional exponents first. It makes the power rule much easier to apply and prevents errors with the chain rule or product rule.