2. Find the first derivative for the following function. \[ f(x)=10 x^{10}-2 \sqrt{x}+3.31 \] \[ f \]

The first derivative of the function \( f(x) = 10x^{10} - 2\sqrt{x} + 3.31 \) is calculated by applying the power rule and the derivative of the square root function. For the term \( 10x^{10} \), the derivative is \( 100x^9 \). The term \( -2\sqrt{x} \) can be rewritten as \( -2x^{1/2} \), and its derivative results in \( -1x^{-1/2} \) or \( -\frac{1}{\sqrt{x}} \). The constant \( 3.31 \) disappears. Thus, the first derivative is: \[ f'(x) = 100x^9 - \frac{1}{\sqrt{x}}. \] This snazzy formula shows how quickly your function is changing at any point along its x-axis! Everything about calculus is based on understanding how functions behave, and that’s super useful in real life. For instance, if you’re designing a roller coaster, knowing how steep the hills will be as the speed changes at different points can make all the difference in safety and excitement! So, next time you hop on a ride, remember there’s a bit of calculus behind the thrills!