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This second derivative calculator finds \(f”(x)\) step by step. It shows the first derivative, then differentiates again. You can also plug in a value of x to see if the graph curves up or down at that point.
How to Use It
- Type your function, for example
x^4 - 3x^2. - Optional: enter a value in Evaluate at x = to get \(f”\) at that point.
- Press Solve. You will see two sets of steps: one for \(f'(x)\) and one for \(f”(x)\).
What Is the Second Derivative?
The second derivative is the derivative of the derivative. We write it as \(f”(x)\) or \(\frac{d^2y}{dx^2}\).
- The first derivative \(f’\) tells you the slope.
- The second derivative \(f”\) tells you how the slope is changing.
In physics, if \(s(t)\) is position, then \(s'(t)\) is velocity and \(s”(t)\) is acceleration.
Concavity: What the Sign of f” Means
| If | The graph is | Shape |
|---|---|---|
| \(f”(x) > 0\) | Concave up | Like a cup ∪ (holds water) |
| \(f”(x) < 0\) | Concave down | Like a frown ∩ |
| \(f”(x) = 0\) and the sign changes | Inflection point | Where the curve switches |
The Second Derivative Test
Use it to tell if a critical point is a high point or a low point.
- Find where \(f'(x) = 0\). These are critical points.
- Plug each one into \(f”(x)\).
- If \(f” > 0\), it is a local minimum. If \(f” < 0\), it is a local maximum. If \(f” = 0\), the test does not decide.
Worked Examples
Example 1: \(f(x) = x^4 – 3x^2\)
\(f'(x) = 4x^3 – 6x\). Then \(f”(x) = 12x^2 – 6\).
At \(x = 0\): \(f”(0) = -6 < 0\), so the graph is concave down there. Since \(f'(0) = 0\), the point \(x = 0\) is a local maximum.
Example 2: \(f(x) = \sin(2x)\)
\(f'(x) = 2\cos(2x)\) (chain rule). Then \(f”(x) = -4\sin(2x)\).
Notice that \(f”(x) = -4f(x)\). Sine and cosine curves always bend back toward the x-axis.
Example 3: \(f(x) = \ln x\)
\(f'(x) = \frac{1}{x}\), so \(f”(x) = -\frac{1}{x^2}\). This is negative for every \(x > 0\). The graph of \(\ln x\) is always concave down.
Example 4: \(f(x) = x^3 e^x\)
Use the product rule twice. \(f'(x) = 3x^2e^x + x^3e^x = e^x(x^3 + 3x^2)\). Then \(f”(x) = e^x(x^3 + 6x^2 + 6x)\).
Common Mistakes
- Not simplifying f’ first. A messy first derivative makes the second step much harder.
- Squaring the first derivative. \(f”\) means “differentiate twice,” not \((f’)^2\).
- Calling every f”=0 point an inflection point. The sign of \(f”\) must change. For \(x^4\), \(f”(0) = 0\) but there is no inflection point.
Learn the Math Behind the Second Derivative Calculator
The second derivative tells you how the slope itself changes: whether a graph bends up or down, and the acceleration of a moving object.
Frequently Asked Questions
Can I get the third or fourth derivative?
Yes. Use the derivative calculator and set the order to 3, 4, or 5.
What does a second derivative of zero mean?
The graph is momentarily straight there. It might be an inflection point, but only if the sign of \(f”\) changes.
How is the second derivative used in real life?
It describes acceleration in physics, and it helps find the best (maximum or minimum) values in business and engineering problems.
Related Guides
- Antiderivative vs derivative
- What is calculus used for in real life?
- How to study calculus on your own