Second Derivative Calculator
Compute $\frac{d^2}{dx^2}$ for any function. Supports polynomials, rational functions, trigonometry, and exponentials.
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Calculation Results
Second Derivative Solution
First Derivative Step
Graph Visualization
Detailed Calculation Steps
Derivation Process
1
Identify the Function
We start with the function $f(x)$:
2
Find the First Derivative
Differentiate $f(x)$ with respect to $x$ to find $f'(x)$:
3
Find the Second Derivative
Differentiate the result from Step 2 ($f'(x)$) once more:
4
Final Answer
Reference: Common Rules
Power Rule
$\frac{d}{dx}(x^n) = nx^{n-1}$
Chain Rule
$\frac{d}{dx}f(g) = f'(g) \cdot g’$
Product Rule
$(uv)’ = u’v + uv’$
Quotient Rule
$(\frac{u}{v})’ = \frac{u’v – uv’}{v^2}$
Professor’s Lecture Notes
The Complete Guide to the Second Derivative: Concavity, Inflection & Acceleration
A definitive guide to understanding $f”(x)$, using the Second Derivative Test for optimization, and analyzing the physics of motion.
In Calculus I, the first derivative ($f’$) gets all the attention. It tells us the slope, the speed, the direction. But to truly understand the behavior of a function—how it bends, twists, and accelerates—we must look deeper.
The Second Derivative ($f”$ or $\frac{d^2y}{dx^2}$) is the “derivative of the derivative.” It measures the rate at which the slope is changing. I developed this free Second Derivative Calculator to help students visualize these abstract concepts, verify their Inflection Points, and master the art of curve sketching.
1. Decoding Concavity: The “Cup” Analogy
The most important application of the second derivative is determining Concavity. This tells us if a graph is curving upwards or downwards.
$$ \text{Concavity} = \text{Sign of } f”(x) $$
Concave Up (Cup)
Holds Water • “Smiling”
$$f”(x) > 0$$
Concave Down (Frown)
Spills Water • “Frowning”
$$f”(x) < 0$$
2. How to Find Inflection Points (Step-by-Step)
An Inflection Point is a dramatic moment in the life of a function: it’s where the concavity flips from up to down (or vice versa). Finding these is a classic exam question.
The 3-Step Verification Process
- Derive Twice: Calculate $f”(x)$ using the Power Rule, Chain Rule, or Product Rule.
- Find Candidates: Solve for $x$ where $f”(x) = 0$ or where $f”(x)$ is undefined. These are potential inflection points.
- The “Sign Test” (Crucial): Pick test numbers on either side of your candidate.
If the sign changes (e.g., $+\to-$), it IS an inflection point.
If the sign stays the same (e.g., $+\to+$), it is NOT an inflection point.
⚠️ The “Flat but not Flipping” Trap
Consider $f(x) = x^4$.
Second derivative: $f”(x) = 12x^2$.
If we set $f”(x)=0$, we get $x=0$. Is this an inflection point? NO!
Test $x=-1$: $f”(-1) = 12 > 0$ (Concave Up).
Test $x=1$: $f”(1) = 12 > 0$ (Concave Up).
Since concavity didn’t change (Up to Up), $x=0$ is just a flat spot, not an inflection point.
3. The Second Derivative Test for Optimization
Why calculate $f”$ if we can find Max/Min with just the first derivative? Because the Second Derivative Test is often faster. It allows you to classify a critical point without making a messy sign chart.
| Critical Point ($f'(c)=0$) | Value of $f”(c)$ | Shape at $c$ | Conclusion |
|---|---|---|---|
| Horizontal Tangent | Positive (+) | Concave Up ($\cup$) | Local Minimum (Valley Bottom) |
| Horizontal Tangent | Negative (-) | Concave Down ($\cap$) | Local Maximum (Hill Top) |
| Horizontal Tangent | Zero (0) | Flat / Unknown | Test Fails (Must use 1st Derivative Test) |
4. Real-World Applications: Physics & Economics
Calculus isn’t just abstract math. The second derivative represents “acceleration” in almost every field of science.
Physics: Motion and Jerk
If $s(t)$ represents the position of a car at time $t$:
- Velocity ($v$): $s'(t)$. How fast you are moving.
- Acceleration ($a$): $s”(t)$. How fast your speed is changing (Pressing the gas pedal).
- Jerk ($j$): $s”'(t)$. The Third Derivative. This is the “snap” you feel when you brake suddenly. Engineering focuses on minimizing jerk to ensure passenger comfort.
Economics: Diminishing Returns
In economics, if a profit function $P(x)$ is increasing ($P’ > 0$) but the second derivative is negative ($P” < 0$), it represents The Law of Diminishing Returns. You are still making money, but every new dollar earns you less than the previous one. The curve is flattening out.
5. Deep Dive Walkthrough: Rational Functions
Polynomials are easy. Let’s tackle a rational function, which requires the Quotient Rule. This is where most students make algebra errors.
Problem: Find $f”(x)$ for $f(x) = \frac{x}{x+1}$
Step 1: First Derivative (Quotient Rule)
$$ f'(x) = \frac{(1)(x+1) – (x)(1)}{(x+1)^2} = \frac{x+1-x}{(x+1)^2} = \frac{1}{(x+1)^2} = (x+1)^{-2} $$
Tip: Rewriting as a negative exponent makes the next step easier!
Step 2: Second Derivative (Chain Rule)
$$ f”(x) = -2(x+1)^{-3} \cdot (1) = \frac{-2}{(x+1)^3} $$
Step 3: Analysis
$f”(x)$ is never zero. However, it is undefined at $x=-1$ (Vertical Asymptote).
For $x > -1$, denominator is positive, so $f” < 0$ (Concave Down).
For $x < -1$, denominator is negative, so $f'' > 0$ (Concave Up).
6. Frequently Asked Questions (FAQ)
Can the second derivative be undefined?
Yes. If the graph has a “cusp” (sharp corner) or a vertical tangent, the second derivative will not exist. These points are also critical values for determining concavity changes.
What is the notation d²y/dx²?
This is Leibniz Notation. It means “differentiate $y$ with respect to $x$, twice.” It is equivalent to $f”(x)$ or $y”$. It helps remind you what variable you are changing.
Does this calculator handle implicit differentiation?
This specific tool is for explicit functions ($y=$). For equations like $x^2+y^2=1$, please use our dedicated Implicit Differentiation Calculator which solves for $y”$ using the implicit formula.
How do I find the interval of concavity?
Use our calculator to find the inflection points (where $f”=0$). These points divide the number line into intervals. Pick a test number in each interval and plug it into $f”(x)$. If positive, that interval is Concave Up.
7. Authoritative References
To ensure accuracy, our calculator’s logic is based on standard theorems found in these textbooks:
Stewart, J. (2015). Calculus: Early Transcendentals (8th Ed).
Chapter 4.3: “How Derivatives Affect the Shape of a Graph”. The gold standard for Concavity definitions.
Chapter 4.3: “How Derivatives Affect the Shape of a Graph”. The gold standard for Concavity definitions.
Paul’s Online Math Notes (Lamar University)
Detailed problem sets for “The Shape of a Graph, Part II”. Excellent for practice.
Visit Paul’s Notes →
Detailed problem sets for “The Shape of a Graph, Part II”. Excellent for practice.
Visit Paul’s Notes →
Analyze Your Function’s Curvature Now
Don’t guess the shape. Get the exact second derivative, inflection points, and concavity intervals instantly.
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— Dr. Math (GoCalc Contributor), PhD in Applied Mathematics.
Making higher-order calculus accessible to everyone.
Making higher-order calculus accessible to everyone.