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Second derivative calculator

Find the second derivative

Differentiate an explicit function twice, with the first derivative kept visible so each rule and simplification can be checked.

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Try an example

Result

Second derivative

d2dx2(x3ex)=ex(x3+6x2+6x)\frac{d^2}{dx^2}(x^3e^x)=e^x(x^3+6x^2+6x)
Conditions
  • x is real.
  • The function and both displayed derivatives exist for every real x.

Steps

  1. Find the first derivative. Apply the product rule to x cubed times e to the x.f′(x)=3x2ex+x3ex=ex(x3+3x2)f'(x)=3x^2e^x+x^3e^x=e^x(x^3+3x^2)
  2. Differentiate the first derivative. Use the product rule again on the factored expression.f′′(x)=ex(x3+3x2)+ex(3x2+6x)f''(x)=e^x(x^3+3x^2)+e^x(3x^2+6x)
  3. Combine like terms. Factor out e to the x and add the polynomial terms.f′′(x)=ex(x3+6x2+6x)f''(x)=e^x(x^3+6x^2+6x)
Independent check

Differentiating the expanded first derivative term by term produces the same second derivative, and centered second differences agree at regular sample points.

Scope

What this second derivative calculator covers

The second derivative is the derivative of the first derivative. It measures how the original rate of change is itself changing and supports concavity and acceleration analysis when the context and domain permit.

Two visible derivative stages

Keep f prime and f double prime separate so a mistake in the first pass can be found before simplification.

Examples: polynomials, products, compositions

Product and quotient functions

Apply the required rule on each pass, since the structure of the first derivative may differ from the original function.

Examples: x squared times e to the x, x over x plus 1

Trigonometric and exponential functions

Track repeated chain-rule factors and signs through two differentiations.

Examples: sine of 3x, e to the 2x

Domain-aware results

Carry the original domain and remove points where either the first or second derivative is undefined.

Examples: natural logarithm, fractional powers, rational functions

How to use it

Enter enough information for one clear task

  1. 1
    Enter one explicit function

    Use parentheses around function arguments and denominators so both derivative passes preserve the intended structure.

  2. 2
    Choose the variable

    Other symbols are treated as constants only when their role is unambiguous.

  3. 3
    Inspect the first derivative

    Check its rules, signs, and domain before using it as the input to the second pass.

  4. 4
    Use the result in context

    For concavity or acceleration, evaluate f double prime only on intervals where the original function and required derivatives exist.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Polynomial

apply the power rule twice

d2dx2x5\frac{d^2}{dx^2}x^5

Expected result

20x320x^3

Trigonometric composition

chain rule twice

d2dx2sin⁡(3x)\frac{d^2}{dx^2}\sin(3x)

Expected result

−9sin⁡(3x)-9\sin(3x)

Exponential composition

repeat the inner scale factor

d2dx2e2x\frac{d^2}{dx^2}e^{2x}

Expected result

4e2x4e^{2x}

Natural logarithm

differentiate one over x

d2dx2ln⁡x\frac{d^2}{dx^2}\ln x

Expected result

−1x2-\frac1{x^2}

Rational function

simplify the first derivative

d2dx2(xx+1)\frac{d^2}{dx^2}\left(\frac{x}{x+1}\right)

Expected result

−2(x+1)3-\frac{2}{(x+1)^3}

Polynomial times exponential

product rule twice

d2dx2(x2ex)\frac{d^2}{dx^2}(x^2e^x)

Expected result

ex(x2+4x+2)e^x(x^2+4x+2)

Complete example

Differentiate x squared times ln x twice

The natural logarithm sets the real domain. The first derivative uses the product rule, and its two terms are then differentiated separately.

f(x)=x2ln⁡x,x>0f(x)=x^2\ln x,\qquad x>0
  1. 1
    Find the first derivative

    Differentiate x squared and ln x in separate product-rule terms.

    f′(x)=2xln⁡x+xf'(x)=2x\ln x+x
  2. 2
    Differentiate 2x ln x

    The product rule gives one term from 2x and one from ln x.

    ddx(2xln⁡x)=2ln⁡x+2\frac{d}{dx}(2x\ln x)=2\ln x+2
  3. 3
    Differentiate the remaining x

    The derivative contributes one more constant.

    ddxx=1\frac{d}{dx}x=1
  4. 4
    Combine and keep the domain

    Add the terms without extending the original logarithm domain.

    f′′(x)=2ln⁡x+3,x>0f''(x)=2\ln x+3,\qquad x>0
d2dx2(x2ln⁡x)=2ln⁡x+3(x>0)\frac{d^2}{dx^2}(x^2\ln x)=2\ln x+3\quad(x>0)

Verification: Differentiating f prime term by term gives 2 ln x + 2 + 1. Centered second differences of the original function agree at positive sample points away from zero.

Avoidable errors

Common mistakes and how to fix them

Skipping the first derivative

Problem: Trying to jump directly from f to f double prime and losing a product or chain factor.

Why it matters: The algebraic structure often changes after the first pass.

Better approach: Write and simplify f prime before differentiating again.

Reusing the first rule mechanically

Problem: Applying the original rule pattern without examining the new expression.

Why it matters: The first derivative can become a sum, product, or quotient with different outer structure.

Better approach: Identify the outermost operation again before the second pass.

Losing a repeated chain factor

Problem: Reporting the second derivative of sine of 3x as minus 3 sine of 3x.

Why it matters: Both derivative passes contribute a factor of 3.

Better approach: Show the inner derivative on each pass, giving minus 9 sine of 3x.

Extending the domain

Problem: Using a simplified f double prime where the original logarithm or denominator was undefined.

Why it matters: Differentiation does not add points to the original function's domain.

Better approach: Intersect the domains of f, f prime, and f double prime.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Differentiate the displayed first derivative independently and compare the simplified expression.
  • Compare centered second differences of the original function at regular interior sample points.
  • Check the original, first-derivative, and second-derivative domains before making a concavity claim.

When to stop and revise the input

  • Nondifferentiable points, corners, cusps, and piecewise joins require interval-by-interval analysis.
  • A zero second derivative at one point does not by itself prove an inflection point; concavity must change across the point.
  • Use the implicit differentiation calculator when the function is given only by an equation in x and y.

Common questions

Find the second derivative FAQ

What is a second derivative?

It is the derivative of the first derivative. It measures how the original rate of change varies with the independent variable.

What does the second derivative say about concavity?

On an interval where f double prime is positive, the graph is concave up; where it is negative, the graph is concave down. Domain and continuity conditions still matter.

Does f double prime equal zero prove an inflection point?

No. It identifies a candidate. The concavity must actually change across the point, and the function must be defined there for the usual graph interpretation.

How can I check a second derivative?

Differentiate the first derivative through a separate simplification path and compare centered second differences of the original function at valid interior points.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.