Differentiate the function. \[ u=\sqrt[7]{t}+2 \sqrt{t^{7}} \] \( u^{\prime}=\square \)

To differentiate the function \( u = \sqrt[7]{t} + 2 \sqrt{t^7} \), we first rewrite it for easier differentiation. We can express \( u \) as: \[ u = t^{1/7} + 2(t^7)^{1/2} = t^{1/7} + 2t^{7/2}. \] Now, differentiate: \[ u' = \frac{d}{dt}\left(t^{1/7}\right) + \frac{d}{dt}\left(2t^{7/2}\right). \] Using the power rule, we get: \[ u' = \frac{1}{7}t^{-6/7} + 2 \cdot \frac{7}{2}t^{5/2} = \frac{1}{7}t^{-6/7} + 7t^{5/2}. \] So, the derivative is: \[ u' = \frac{1}{7}t^{-6/7} + 7t^{5/2}. \] ✨ Remember, the power rule is your best friend when working with derivatives! Whenever you see a term that looks like \( t^n \), just bring down the exponent and decrease it by one. Fun fact: this rule applies to any real number exponent, not just integers! 🎉 🌍 In real-world applications, differentiation is crucial in fields like physics and economics. For instance, understanding how speed (the derivative of distance) changes over time can help in planning safer road systems. So whether you’re optimizing a business model or predicting the trajectory of a rocket, derivatives are your ticket to success! 🚀