Question

What’s the difference between the rates of change in the cube root:

y = cuberoot x + 2

Answer 1

To find the rate of change of the function \( y = \sqrt[3]{x} + 2 \), we need to find its derivative, which gives us the rate of change of \( y \) with respect to \( x \).

1. **Differentiate the function**:

The function can be rewritten in exponent form:
\[
y = x^{1/3} + 2
\]
Now, we can differentiate it:
\[
\frac{dy}{dx} = \frac{d}{dx}(x^{1/3}) + \frac{d}{dx}(2) = \frac{1}{3} x^{-2/3} + 0 = \frac{1}{3} \cdot \frac{1}{\sqrt[3]{x^2}}
\]
Therefore, the derivative simplifies to:
\[
\frac{dy}{dx} = \frac{1}{3\sqrt[3]{x^2}}
\]

2. **Interpret the result**:

The expression \( \frac{1}{3\sqrt[3]{x^2}} \) tells us how the rate of change of \( y \) depends on \( x \). Specifically, as \( x \) increases, the cube root of \( x^2 \) also increases, causing \( \frac{dy}{dx} \) to decrease. This indicates that the rate of change of \( y \) gets smaller as \( x \) gets larger.

3. **Summary**:

- The function \( y = \sqrt[3]{x} + 2 \) has a derivative of \( \frac{dy}{dx} = \frac{1}{3\sqrt[3]{x^2}} \).
- The rate of change decreases as \( x \) increases.
- The cube root function has a continuously positive slope for all \( x \), making \( y \) an increasing function overall.

If you have particular values of \( x \) in mind, we can evaluate \( \frac{dy}{dx} \) at those points for a clearer understanding of the change rate at specific intervals.