Second Derivative Calculator

This second derivative calculator finds \( f'(x) \) and \( f''(x) \) exactly, simplifies both, and evaluates them at any point you choose. Use it to check concavity, run the second derivative test at a critical point, or get acceleration from a position function, without differentiating twice by hand.

Everything is free and runs in your browser, with no sign-up.

How to use the second derivative calculator

1. Enter \( f(x) \) in the first box. Use ^ for powers, sqrt(x), sin(x), cos(x), tan(x), ln(x), e^(x) and pi. Implicit multiplication such as 3x or x^2 ln(x) works.
2. Enter the point where you want the derivatives evaluated in “Evaluate at x”. It accepts expressions like pi/4.
3. Press Differentiate. You get four rows: \( f'(x) \), \( f''(x) \), \( f'(x_0) \) and \( f''(x_0) \). Values are shown in exact form (like \( \sqrt2 \) or \( \frac{\pi}{2} \)) when the calculator recognizes one, with a decimal alongside.

Formula

The second derivative is the derivative of the derivative:

$$f''(x) = \frac{d}{dx}\left[f'(x)\right] = \frac{d^2 f}{dx^2}$$

The calculator differentiates your function symbolically, simplifies \( f'(x) \), then differentiates that result again. The values at your point come from plugging \( x_0 \) into the two exact formulas, not from a numerical approximation.

What the sign of \( f'' \) tells you:

• \( f''(x_0) > 0 \): the graph is concave up at \( x_0 \) (bending upward, like a cup).
• \( f''(x_0) < 0 \): concave down (like a cap).
• \( f''(x_0) = 0 \): no conclusion on its own; check whether \( f'' \) changes sign.

Worked example

Example 1. Find \( f' \) and \( f'' \) for \( f(x) = x^2\ln x \), and evaluate them at \( x = 1 \).

The product rule gives

$$f'(x) = 2x\ln x + x^2\cdot\frac1x = 2x\ln x + x$$

Differentiate again, using the product rule on \( 2x\ln x \):

$$f''(x) = 2\ln x + 2 + 1 = 2\ln x + 3$$

At \( x = 1 \), \( \ln 1 = 0 \), so \( f'(1) = 1 \) and \( f''(1) = 3 \). The function is increasing and concave up there. These are the values the calculator above shows with its default input.

Example 2. For \( f(x) = \tan x \) at \( x = \frac{\pi}{4} \): \( f'(x) = \sec^2 x \) and, by the chain rule, \( f''(x) = 2\sec^2 x\tan x \). Since \( \sec^2\frac{\pi}{4} = 2 \) and \( \tan\frac{\pi}{4} = 1 \), you get \( f'(\frac{\pi}{4}) = 2 \) and \( f''(\frac{\pi}{4}) = 4 \). Type tan(x) and pi/4 to check. (See the derivative of tan(x) for where \( \sec^2 x \) comes from.)

What to do with the second derivative

• Second derivative test. At a critical point where \( f'(c) = 0 \), a positive \( f''(c) \) means a local minimum and a negative one means a local maximum. The full procedure is in how to find critical points.
• Inflection points. Set \( f''(x) = 0 \) and check for a sign change. In Example 1, \( 2\ln x + 3 = 0 \) at \( x = e^{-3/2} \approx 0.2231 \), and \( f'' \) goes from negative to positive there, so the graph switches from concave down to concave up. More in how to find inflection points, or let the inflection point calculator find them on an interval.
• Motion. If \( f \) gives position over time, \( f' \) is velocity and \( f'' \) is acceleration.

A common trap: \( f''(x_0) = 0 \) does not guarantee an inflection point. For \( f(x) = x^4 \), \( f''(x) = 12x^2 \) is zero at 0 but never negative, so the graph is concave up on both sides.

Related guides

• How to Find Inflection Points and Concavity
• How to Find Critical Points
• Derivative of tan(x)
• Product Rule

Further reading

• Higher Order Derivatives (Paul’s Online Math Notes) — second, third and higher derivatives with worked examples.
• The Shape of a Graph, Part II (Paul’s Online Math Notes) — concavity and the second derivative test.

FAQ

How do you find the second derivative?

Differentiate the function once to get \( f'(x) \), simplify, then differentiate \( f'(x) \) again. The calculator does both steps and shows each result.

What does a negative second derivative mean?

The graph is concave down at that point: the slope is decreasing. At a critical point, a negative second derivative means a local maximum.

Can I evaluate the second derivative at a specific point?

Yes. Enter the point in “Evaluate at x” (numbers or expressions like pi/3) and the calculator shows both \( f'(x_0) \) and \( f''(x_0) \).

Does it show the steps?

This tool shows the two exact derivatives and their values. For a rule-by-rule breakdown, use the step-by-step derivative calculator and set the order to “Second derivative”.