double derivative calculator

What Is a Double Derivative?

The double derivative (also called the second derivative) measures how the first derivative changes. If the first derivative tells you slope, the second derivative tells you how that slope is bending. In notation, if your function is f(x), then:

First derivative: f'(x)
Second derivative: f''(x)

Why the Second Derivative Matters

The second derivative appears across math, engineering, economics, and physics. It helps you:

• Determine concavity (concave up or concave down).
• Find local maxima and minima with the second derivative test.
• Model acceleration when position is a function of time.
• Understand curvature and optimization behavior in real systems.

How to Use This Double Derivative Calculator

Step 1: Enter your function

Type a function in terms of your variable (usually x). For example: x^4 - 3x^2 + 2x + 1.

Step 2: Pick the variable

The default variable is x. If your function uses a different variable (like t), enter that instead.

Step 3: (Optional) Enter a point

Provide a value to evaluate the second derivative numerically, such as x = 2.

Step 4: Click calculate

You’ll get the symbolic second derivative and, when a point is supplied, the computed value at that point.

Examples

Example 1: Polynomial

If f(x) = x³, then f'(x) = 3x² and f''(x) = 6x. At x = 2, f''(2) = 12.

Example 2: Trigonometric

If f(x) = sin(x), then f'(x) = cos(x) and f''(x) = -sin(x).

Example 3: Exponential + trig product

For f(x) = e^xcos(x), the second derivative is a combined expression from product rule applications. This tool handles that automatically.

Concavity and the Second Derivative Test

• If f''(a) > 0, the graph is concave up near a (local minimum candidate).
• If f''(a) < 0, the graph is concave down near a (local maximum candidate).
• If f''(a) = 0, the test is inconclusive; check higher derivatives or graph behavior.

Input Tips

• Use ^ for powers: x^2
• Use explicit multiplication when needed: 3*x, x*sin(x)
• Natural log is log(x)
• Square root is sqrt(x)

Frequently Asked Questions

Is this symbolic or numerical?

The second derivative is computed symbolically, then optionally evaluated numerically at a point.

Can I use decimals?

Yes. You can enter decimal constants and decimal evaluation points.

What if my expression fails?

Double-check syntax, variable names, and parentheses. Use explicit multiplication where needed.