When differentiating functions that involve powers of variables, such as \( r^2 \), apply the power rule: \( \frac{d}{dr}(r^n) = n r^{n-1} \). This simplifies the process significantly. Also, remember to substitute numerical values carefully and check cube roots or other roots to ensure the calculation is correct before substituting them into your derivative expression.
The formula for the total surface area \( S \) of a hemisphere is \( S = 3\pi r^2 \). Differentiating \( S \) with respect to \( r \) yields \( \frac{dS}{dr} = 6\pi r \). Given \( r = \sqrt[3]{1.331} \), we first calculate \( r \). Since \( 1.1^3 = 1.331 \), \( r = 1.1 \). Substituting this value into the derivative, we get \( \frac{dS}{dr} = 6\pi (1.1) = 6.6\pi \). Therefore, the rate of change of the total surface area with respect to the radius is \( 6.6\pi \).