I was introduced to calculus a few weeks ago, and while I can "solve" problems consisting of derivatives and integrals, I still do not truly understand what the derivative means.
Here are some of the arguments/explanations I have heard :
1. Derivative is the instantaneous rate of change.
However, this to me never makes sense, for a change to occur there needs to be an interval, but if there is an interval, then it is not instantaneous?
2. Derivative is the slope of the tangent line.
This is simple and easy to understand, but it I do not understand how the "slope" of the tangent line tells us how "fast" the function is changing at a point.
3. Derivative is the sensitivity of the function at the point.
This to me, is the most appealing definition, at a point, the derivative measures how much my function will change around that point if i make a tiny change in my input variable. However, this still causes a bit of confusion to me. How does this "sensitivity" intuition lead to the limit definition of the derivative?
I am sorry, if I made any conceptual errors in my interpretations of above definitions. It would be a great help if someone could help me understand derivatives better.
Best Answer
Since you like "sensitivity" the best, one way to think of the derivative is "how much the function stretches or shrinks a small interval." Suppose the derivative of $f(x)$ at $x=x_0$ is $1/3$. If you take a small interval $(x_0-\delta,x_0+\delta)$ which has width $2\delta$ and put all those values into the function, then their image (on the $y$-axis) is another small interval. The width of that interval will be about $(1/3)2\delta.$ The smaller the interval on the $x$-axis, the closer the width of the interval on the $x$-axis will be to exactly $1/3.$ The derivative tells you that little neighborhoods of $x_0$ get shrunk by a factor of $1/3$ when you push them through $f$.