what is the derivative of $$ {\displaylines16\cdot\text{ third root (x}^2)\\ } $$

The given expression is $ 16 \cdot \sqrt[3]{x^2} $, which is equivalent to $ 16 \cdot (x^2)^{1/3} $.

To find the derivative of this expression with respect to $ x $, follow these steps:

1. Rewrite the expression in a form suitable for differentiation:
   \[
   y = 16 \cdot (x^2)^{1/3}
   \]

2. Use the chain rule. First, apply the power rule to differentiate $ (x^2)^{1/3} $.

   \[
   \frac{d}{dx} \left( (x^2)^{1/3} \right) = \frac{1}{3} \cdot (x^2)^{-2/3} \cdot 2x
   \]
   Simplifying:
   \[
   \frac{d}{dx} \left( (x^2)^{1/3} \right) = \frac{2x}{3} \cdot (x^2)^{-2/3}
   \]

3. Now multiply this result by 16:
   \[
   \frac{d}{dx} \left( 16 \cdot (x^2)^{1/3} \right) = 16 \cdot \frac{2x}{3} \cdot (x^2)^{-2/3}
   \]

4. Simplify:
   \[
   \frac{d}{dx} = \frac{32x}{3} \cdot (x^2)^{-2/3}
   \]
   Or equivalently:
   \[
   \frac{d}{dx} = \frac{32x}{3x^{4/3}} = \frac{32}{3x^{1/3}}
   \]

Thus, the derivative of $ 16 \cdot \sqrt[3]{x^2} $ is:
\[
\frac{32}{3x^{1/3}}
\]

Would you like further explanation or details on any step?

Here are 5 related questions:
1. What is the derivative of $ \sqrt[3]{x^3} $?
2. How do you find the derivative of $ x^n $ for any power $ n $?
3. What is the general rule for the chain rule in calculus?
4. How do you compute higher-order derivatives for power functions?
5. What is the derivative of a product of functions like $ x \cdot \sqrt[3]{x^2} $?

**Tip:** When differentiating functions with exponents, it's helpful to rewrite them in power form to apply the power rule more easily.

what is the derivative of $$ 16 \cdot \text{ third root }(x^2) $$

Discover how to find the derivative of 16 times the cube root of x^2 with step-by-step instructions. This solution uses key calculus concepts, including the chain rule and power rule, ideal for students in Grades 11-12.

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