Calculus gets truly exciting when we explore how functions
change — and derivatives are the heart of this adventure.
In this post, we’ll find and visualize the first four derivatives of 
using SageMath.
Not only will we calculate them symbolically, but we’ll also bring them to life
with colorful plots!
1. π Understanding the
Function
Our function 
blends logarithmic and
quadratic behavior.
It’s smooth, continuous, and defined everywhere on 
.
Here’s our roadmap:
- Find
the first derivative
(how fast 
changes).
- Find
the second derivative
(how the curve bends).
- Find
the third and fourth derivatives
(to capture subtle twists).
2. π Step-by-Step
Differentiation with SageMath
Differentiating by hand can get tedious, but SageMath
makes it quick and reliable.
Here’s the simple code:
𧩠Quick Insights:
reflects how steeply climbs or falls.- tells us about bumps and dips
(concavity).
capture
even finer ripples in the graph.
3. π¨ Visualizing the
Function and Derivatives
A picture is worth a thousand calculations!
Let’s plot f(t) and its derivatives together:
π️ Annotated
Observations:
is smooth and grows steadily for 
.
dips briefly at 
, showing a small pause
in the function's climb.
and higher derivatives introduce waves,
revealing more intricate behavior.
4. π Interpreting the
Results
πΆ First Derivative :
- Tells
how steep the curve is at any point.
- Positive
slope? Curve goes up!
- Negative
slope? Curve drops!
π Second Derivative :
- Checks
if the graph is smiling (concave up) or frowning (concave down).
- Helps
us locate inflection points where the curve changes its mood.
π Higher Derivatives :
- Capture
finer "ripples" in the curve.
- Useful
in physics, engineering, and deep modeling where precision matters.
π§ Mini-Challenge for You!
Try differentiating these related functions:
Compare how the presence of polynomials or trigonometric
terms changes their behavior!
π Wrapping Up
We saw how SageMath can transform complex
differentiation into an elegant, visual story.
By plotting f(t) and its derivatives, we understood growth, curvature,
and fine structure — way beyond just formulas.
π What's Next?
Get ready for our next adventure: Implicit Differentiation and cool
applications like verifying the Mean Value Theorem!
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