What does exact and approximation of derivative mean?

Course Q&A Math for Machine Learning Calculus for Machine Learning and Data Science
1

In the “Approximation of derivatives” reading section, as we reduce the area of the triangle formed in the upper plot, the curve for the approximate derivative aligns completely with the exact derivative.

How is the exact derivative curve plotted? Is it computed analytically based on the function that generated the curve on the upper plot?

Through this exercise, What is the point to be taken? I always thought the derivative is the slope i.e., rise over run. But it seems from this exercise that rise over run is an approximation to compute exact derivative. Can you please help clear my understanding by differentiating between exact derivative and this approximation?

Thanks in advance!

2

I think I get it. Derivative is the slope of the tangent. Hence, as we reduce the area of the triangle formed in the upper plot to essentially zero (a point), the line becomes a tangent and its slope is exactly the derivative as per the definition.