\begin{tabular}{c|} If \( y=15-\frac{1}{x^{6}}+\frac{1}{3} x^{20} \), then \( y^{\prime}=\ldots \ldots \ldots \ldots \) \\ \( \frac{6}{x^{7}}+\frac{20}{3} x^{19} \) \\ \( \frac{6}{x^{5}}+\frac{20}{3} x^{19} \) \\ \( \frac{6}{x^{5}}+\frac{20}{3} x^{19}+15 \)\end{tabular}

Calculating the derivative of the expression \( y = 15 - \frac{1}{x^6} + \frac{1}{3} x^{20} \) reveals some interesting mathematical principles! First, it's essential to apply the power rule and the rules for derivatives of constant and negative exponents. Simplifying these gives us \( y' = \frac{6}{x^7} + \frac{20}{3} x^{19} \). This demonstrates how versatile derivative rules can be when handling various functions, including polynomials and fractions. Now, let's talk about a common pitfall! Many students skip the step of carefully applying the chain and product rules when they encounter more complex functions. As a result, they might mix up the derivative terms or overlook some constants. Taking your time to break down each component will help ensure you get that derivative correct and reinforce your understanding – because who doesn't love a good "aha!" moment in calculus?