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 \( x \).
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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: \[ \frac{dy}{dx} = \frac{1}{3\sqrt[3]{x^2}} \]
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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.
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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.
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