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Higher derivatives

How to Find a Second Derivative Step by Step

A second derivative measures how slope changes. Compute it from the complete first derivative, not by skipping directly from the original function.

Direct answer

f″(x) = d/dx [f′(x)]

The first derivative must itself be differentiable on the interval.

When to use how to find a second derivative

Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.

Ask first 1

Which structure controls the method?

What to do

Use a second derivative when the task asks how slope changes rather than the slope itself.

Ask first 2

Which condition can change the answer?

What to do

The first derivative must itself be differentiable on the interval.

Ask first 3

How should the result be checked?

What to do

Show two separate derivative passes and set calculator order to two.

Two visible derivative passes

Compute f′(x) and retain its complete domain.

Treat that entire result as a new differentiation problem.

The second pass may introduce product or chain rules even when the first looked simple.

Interpret concavity only where f″ exists.

1
f x=x3−2 xf\,x = x^{3} - 2\,x
2
f'(x)=3x²−2
3
f''(x)=6x

Worked examples

d2dx2[x4]\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left[x^{4}\right]
12 x212\,x^{2}

Apply the power rule twice.

d2dx2[sin⁡x]\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left[\sin x\right]
−sin⁡x-\sin x

Sine becomes cosine, then negative sine.

d2dx2[e2 x]\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left[e^{2\,x}\right]
4 e2 x4\,e^{2\,x}

Each pass contributes a factor 2.

Common mistakes

  • • Stopping at f′.
  • • Dropping terms from f′.
  • • Missing a repeated inner factor.
  • • Discussing curvature where f″ fails.

Error clinic: locate the first invalid step

Incorrect attempt

[x⁴]''=4x³

Why it fails

This is only the first derivative.

Correction

[x⁴]''=12x²

Incorrect attempt

[exp(2x)]''=2exp(2x)

Why it fails

The second pass contributes another factor 2.

Correction

[exp(2x)]''=4exp(2x)

Practice the same idea with a changed structure

Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.

d2dx2[sin⁡x]\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left[\sin x\right]

Check before solving

Sine becomes cosine, then negative sine.

Reveal the reference answer

Reference answer

−sin⁡x-\sin x
d2dx2[e2 x]\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left[e^{2\,x}\right]

Check before solving

Each pass contributes a factor 2.

Reveal the reference answer

Reference answer

4 e2 x4\,e^{2\,x}

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

求导:x4x^{4}求导:sin⁡x\sin x
Open the calculator without a preset: 求导

Frequently asked questions

Is f″ the square of f′?+

No. It means differentiate f′ once more.

What does f″>0 mean?+

Where the derivatives exist, it indicates the graph is concave up.

Can f′ exist when f″ does not?+

Yes. A first derivative need not itself be differentiable.

Conclusion

Find the complete first derivative, differentiate it again, and interpret curvature only where the second derivative exists.

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