# Second Derivative Calculator

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## Second derivative calculator with steps

Second derivative calculator differentiates the function two times with respect to the corresponding variable. This second differential calculator provides a step-by-step solution. It differentiates the first derivative of the function and then the second derivative.

## How does this second differentiation calculator work?

Follow the below steps to find the 2nd order differential of the function.

• Input the function.
• Select the variable.
• Click the calculate button.
• To enter a new function, press the clear button.

## What is the second derivative?

The second derivative of a function is the derivative of the derivative of that function. It is denoted by f’’(x) or d2f(x)/dx2.
It is usually used to check whether the slope of the tangent line is increasing or decreasing. The first derivative will be increasing if the second derivative is positive. While if the second derivative is negative, then the first derivative will be decreasing.

## How to calculate the second derivative?

Below is a solved example of a second derivative.

Example

Find the second derivative of xsin(x) + 2 with respect to “x”?

Solution

Step 1: First of all, find the first derivative of the function.

$$\frac{d}{dx}\left(xsin\left(x\right)+2\right)=\frac{d}{dx}\left(xsin\left(x\right)\right)+\frac{d}{dx}\left(2\right)$$

$$\frac{d}{dx}\left(xsin\left(x\right)+2\right)=\left[sin\left(x\right)\frac{d}{dx}\left(x\right)+x\frac{d}{dx}\left(sin\left(x\right)\right)\right]+\frac{d}{dx}\left(2\right)$$

$$\frac{d}{dx}\left(xsin\left(x\right)+2\right)=\left[sin\left(x\right)\left(1\right)+x\left(cos\left(x\right)\right)\right]+0$$

$$\frac{d}{dx}\left(xsin\left(x\right)+2\right)=sin\left(x\right)+xcos\left(x\right)$$

Step 2: To find the second derivative, take the derivative of the derivative.

$$\frac{d}{dx}\left[\frac{d}{dx}\left(xsin\left(x\right)+2\right)\right]=\frac{d}{dx}\left(sin\left(x\right)+xcos\left(x\right)\right)$$

$$\frac{d^2}{dx^2}\left(xsin\left(x\right)+2\right)=\frac{d}{dx}sin\left(x\right)+\frac{d}{dx}xcos\left(x\right)$$

$$\frac{d^2}{dx^2}\left(xsin\left(x\right)+2\right)=\frac{d}{dx}sin\left(x\right)+cos\left(x\right)\frac{d}{dx}x+x\frac{d}{dx}cos\left(x\right)$$

$$\frac{d^2}{dx^2}\left(xsin\left(x\right)+2\right)=cos\left(x\right)+cos\left(x\right)\left(1\right)+x\left(-sin\left(x\right)\right)$$

$$\frac{d^2}{dx^2}\left(xsin\left(x\right)+2\right)=cos\left(x\right)+cos\left(x\right)-xsin\left(x\right)$$

$$\frac{d^2}{dx^2}\left(xsin\left(x\right)+2\right)=2cos\left(x\right)-xsin\left(x\right)$$

### References

Example of the second derivative. Study.com | Take Online Courses. Earn College Credit.