Being differentiable is important because it helps us understand how things change and behave in the real world. For example, imagine you're a engineer designing a roller coaster - you want to know how the slope of the track changes at any given point so that you can ensure a smooth ride. If the function representing the track's slope is differentiable, you can use calculus to figure out the rate of change and make adjustments accordingly.
In daily life, we often encounter situations where being differentiable makes a big difference. Think about a basketball player trying to make a shot - they need to be able to adjust their trajectory smoothly to make the shot. If their movement is jerky or erratic, they'll likely miss. In this case, being differentiable means being able to make smooth adjustments to their trajectory, increasing their chances of scoring.
Another example is in finance - imagine you're an investor trying to understand how a stock's price is changing over time. If the function representing the stock's price is differentiable, you can use calculus to analyze the rate of change and make informed investment decisions. It's like having a crystal ball that shows you how the stock's price will change, helping you make smart investments.
11X1 T09 08 implicit differentiation (2010)
