There is a relation between the tangent to a curve of a function and the first derivative of that function. However, how do I show that connection? How can you explain it to someone so that it becomes clear and understandable?
[Math] Relation between the tangent to a curve and the first derivative of a function
calculus
Best Answer
My answer is going to be very informal but geared toward gaining some intuition. I'm assuming "nice curves" and drawing lots of pictures:
Remember that the derivative at $x$ is essentially defined as the limit of the slopes of secant lines near $x$.
This should provide some understanding of the relation you're talking about.
Draw a line through $f(x)$ and $f(x_1)$ where $x_1$ is "close to" $x$.
Then draw a line through $f(x)$ and $f(x_2)$ where $x_2$ is halfway between $x$ and $x_1$.
Then do this with $x_3$ halfway between $x$ and $x_2$, and so on.
Since you're drawing lines that intersect the curve at points $x$ and $x_n$, but $x_n$ keeps getting closer to $x$, "eventually" the line only intersects the curve at $x$.
Alternatively, notice that if you draw a line through $f(x)$ and a nearby $f(x')$, then via some sort of intermediate value argument, if the curve didn't have constant slope between those points, it must have gone "up" or "down" relative to your line between those points, and your line must not really be approximating the local slope of the curve.