The Slope of The Slope: Second Derivatives
The second derivative measures a function's rate of curvature change, obtained by differentiating the first derivative. It's pivotal in determining concavity, inflection points, and optimizing real-world scenarios like motion dynamics.

The second derivatives create their own graph, which is distinct from the original function’s graph. This graph, often called the “second derivative graph” or “concavity graph,” visually represents the rate at which the slope of the original function changes. It helps in understanding the curvature behavior of the original function, indicating where it’s concave up or down, and locating any inflection points where the concavity changes.
Since you are already equipped with the tools required to solve the problems involving the second derivative,
let’s dive into problems:
\( \text{Find the first and second derivatives of } f(x) = x \sin x. \)
\( u = x, \ u’ = 1 \)
\( v = \sin x, \ v’ = \cos x \)
\( f'(x) = 1 \cdot \sin x + x \cdot \cos x \)
\( f'(x) = \sin x + x \cos x \)
\( (\sin x)’ = \cos x \)
\( (x \cos x)’ = 1 \cdot \cos x + x \cdot (-\sin x) \)
\( (x \cos x)’ = \cos x – x \sin x \)
\( f”(x) = \cos x + \cos x – x \sin x \)
\( f”(x) = 2\cos x – x \sin x \)
And that’s the second derivative of \( xsin x \)
Related to This Article
More math articles
- Top 10 Free Websites for PERT Math Preparation
- PSAT / NMSQT Math-Test Day Tips
- How to Identify Relations and Functions? (+FREE Worksheet!)
- How to Prepare for the PERT Math Test?
- HiSET Math FREE Sample Practice Questions
- 6th Grade GMAS Math Worksheets: FREE & Printable
- 5 Best Programmable Calculators
- Top Tablets for Online Math Teaching
- How to Understand Convergent and Divergent Series
- Decoding Magnitudes: A Deep Dive into the Intricacies of Absolute Value Properties
What people say about "The Slope of The Slope: Second Derivatives - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.