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Second Derivative Calculator

Enter the function and select the variable for which the tool will compute its derivative up to the second order, with detailed calculations displayed.

Second Derivative of Please write e^x as e^{x}:

Derivative W.R.T:

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Online Second Derivative Calculator

Our online second derivative calculator helps you compute the second-order derivative of any function quickly and accurately. This tool saves time, reduces errors, and is ideal for students, engineers, and researchers. Learn how to calculate second derivatives using power, product, and chain rules effectively.

What is a Second Derivative?

The second derivative is the derivative of a derivative. While the first derivative measures the rate of change (slope) of a function, the second derivative measures how that rate of change itself changes. In simple terms, it shows the concavity of a function.

Example in physics: The second derivative of position with respect to time gives acceleration:

$$ a = \frac{d(v)}{dt} = \frac{d^2 x}{dt^2} $$

a = acceleration
t = time
x = position
v = velocity
d = instantaneous change

For first derivatives, use our Online Derivative Calculator.

Power Rule for Second Derivative

For x^n, applying the power rule twice gives:

$$ \frac{d^2}{dx^2}[x^n] = \frac{d}{dx} \Big(\frac{d}{dx}[x^n]\Big) = \frac{d}{dx}[n x^{n-1}] = n (n - 1) x^{n-2} $$

Notation for Second Derivative

The second derivative of f(x) can be written as:

f''(x) = (f'(x))'
Or in Leibniz notation: d²y / dx²

Formally:

$$ \frac{d^2y}{dx^2} = \frac{d}{dx}\Big(\frac{dy}{dx}\Big) $$

How to Calculate the Second Derivative

Second derivatives can be calculated using power, product, and chain rules.

Example:

Find the second derivative of:

$$ \frac{d^2}{dx^2} \big[ \sin(x) \cos^3(x) \big] $$

Step 1: Apply the Product Rule

$$ \frac{d}{dx}[f(x) g(x)] = f(x) \frac{d}{dx}g(x) + g(x) \frac{d}{dx}f(x) $$

Let f(x) = cos^3(x), g(x) = sin(x)

Step 2: Apply the Chain Rule

For f(x) = cos^3(x):

Let u = cos(x)
Power rule: u³ → 3u²
Multiply by derivative of u: d/dx[cos(x)] = -sin(x)

Result: -3 sin(x) cos²(x)

For g(x) = sin(x), derivative: d/dx[sin(x)] = cos(x)

Combine and simplify using the product rule and continue applying the chain rule. Final simplified second derivative:

$$ -\sin(2x) - 2 \sin(4x) $$

For integrals, use the Online Integral Calculator.

Second Derivative in Monotony Tables

The second derivative is useful for analyzing the slope and concavity:

Positive second derivative → slope increasing → convex function
Negative second derivative → slope decreasing → concave function
Zero second derivative → slope constant → straight line

How the Second Derivative Calculator Works

Input:

Enter the function to differentiate
Select the variable
Click "Calculate Second Derivative"

Output:

Displays the second derivative with step-by-step simplification

FAQs

What is the second derivative test used for?

It identifies local extrema. If f'(x)=0 and f''(x) > 0, the function has a local minimum. If f''(x) < 0, there is a local maximum.

When is the second derivative positive?

If f''(x) > 0 on an interval, the function is concave upward and its slope is increasing.

Conclusion

Use this online second derivative calculator to perform second-order differentiation accurately and efficiently. It saves time and provides detailed step-by-step results for any function.

References

Wikipedia: Second Derivative, Concavity, Inflection Points, Second Derivative Test
LibreTexts: Concavity, Points of Inflection
ITCC Online: The Second Derivative Test

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