The differential quotient
The differential quotient is the slope of the tangent to the graph of the function, it measures how "fast" the function "changes". Formally it is defined as the following limit:
The animation shows how the slope of the tangent changes.
The Taylor series is a polynomial approximation to infinitely many times differentiable functions.
Truncated Taylor series are often used in numerical analysis to approximate functions. The animation below lets you appreciate how the Taylor approximation gets better when more and more terms containing higher-order derivatives are included.
Many mechanistic models rely on differential equations to model cause-effect relationships. Consult the CoCalc differential equation primer (PDF) on how to solve differential equations in SAGE.