10. Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (scalar multiple, sum/difference, quotient, etc.) you are using in your work. \[ h(t)=\sqrt[5]{t^{4}}+\frac{1}{t^{5}} \]

To find the derivative of the function \( h(t) = \sqrt[5]{t^{4}} + \frac{1}{t^{5}} \), we can first rewrite it in a form that's easier to differentiate. We can express \( \sqrt[5]{t^{4}} \) as \( t^{4/5} \) and \( \frac{1}{t^{5}} \) as \( t^{-5} \). Thus, we have: \[ h(t) = t^{4/5} + t^{-5} \] Now, let's differentiate term by term. For the first term, we use the power rule, which states that \( \frac{d}{dt}[t^n] = n \cdot t^{n-1} \). Applying this, the derivative of \( t^{4/5} \) is: \[ \frac{d}{dt}[t^{4/5}] = \frac{4}{5}t^{(4/5) - 1} = \frac{4}{5}t^{-1/5} \] For the second term \( t^{-5} \), we again apply the power rule: \[ \frac{d}{dt}[t^{-5}] = -5t^{-6} \] Now, combining these results, the derivative \( h'(t) \) is: \[ h'(t) = \frac{4}{5}t^{-1/5} - 5t^{-6} \] In more compact form, we may want to express this with positive exponents if needed, but this derivative captures the necessary rules. To provide a little historical context, calculus emerged as a fundamental area of mathematics in the late 17th century, with contributions from great minds like Isaac Newton and Gottfried Wilhelm Leibniz. Its methods involving derivatives and integrals revolutionized how we approach motion, growth, and change in nature, paving the way for advancements in physics and engineering. In terms of real-world applications, derivatives have a crucial role in various fields, especially in physics for calculating rates of change, such as velocity and acceleration. In economics, derivatives help in finding marginal costs and revenues, which enables businesses to make informed decisions based on how changing one variable affects overall profit. This highlights just how potent a tool the derivative truly is!