Second derivative calculator with steps
Second derivative calculator is used to calculate the 2nd-order differential of the function. It calculates the second-order derivative by differentiating the function twice.
A double derivative calculator provides a step-by-step solution. It calculates the first derivative and then the second derivative of the functions.
How does 2nd derivative calculator work?
Follow the below steps to use this second derivative calculator.
- Enter the function.
- Select the variable.
- Use the keypad icon
to enter the mathematical symbols. - Press the calculate button.
- Press the show more button to view the step-by-step solution.
- Click the reset button to solve another function.
What is the second derivative?
The process of finding the differential of a function with respect to its variable is known as differentiation.
Similarly, the process of finding the second-order derivative of a similar function is known as a second differentiation. It is often called a second-order derivative.
The second-order derivative is a technique that calculates the derivative of the first differential function.
The notation for the second differential is \(\frac{d^2}{dx^2}\) or \(f''\left(x\right)\).
How to calculate the second derivative?
Following are a few examples solved by our second derivative test calculator.
Example 1
Find the second derivative of \(4x^2+cos\left(x\right)\)?
Solution
Step 1: Apply the differential notation on the given function.
\(\frac{d}{dx}\left(4x^2+cos\left(x\right)\right)\)
Step 2: According to sum rule, apply the differential notation separately.
\(\frac{d}{dx}\left(4x^2+cos\left(x\right)\right)=\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(cos\left(x\right)\right)\)
Step 3: Now apply the power rule and differentiate the above term.
\(\frac{d}{dx}\left(4x^2+cos\left(x\right)\right)=\left(4\cdot 2\right)x^{2-1}+\left(-sin\left(x\right)\right)\)
\(\frac{d}{dx}\left(4x^2+cos\left(x\right)\right)=8x-sin\left(x\right)\)
Step 4: Now apply the differential again on the above first derivative function to get the second order differential.
\(\frac{d}{dx}\left[\frac{d}{dx}\left(4x^2+cos\left(x\right)\right)\right]=\frac{d}{dx}\left[8x-sin\left(x\right)\right]\)
\(\frac{d^2}{dx^2}\left(4x^2+cos\left(x\right)\right)=\frac{d}{dx}\left[8x-sin\left(x\right)\right]\)
Step 5: Apply the difference rule and differentiate the function.
\(\frac{d^2}{dx^2}\left(4x^2+cos\left(x\right)\right)=\frac{d}{dx}\left(8x\right)-\frac{d}{dx}\left(sin\left(x\right)\right)\)
\(\frac{d^2}{dx^2}\left(4x^2+cos\left(x\right)\right)=\left(8\left(1\right)\right)-\left(cos\left(x\right)\right)\)
\(\frac{d^2}{dx^2}\left(4x^2+cos\left(x\right)\right)=8-cos\left(x\right)\)
Example 2
Find the second derivative of \(xsin\left(x\right)+32\)?
Solution
Step 1: Apply the differential notation on the given function.
\(\frac{d}{dx}\left(xsin\left(x\right)+32\right)\)
Step 2: According to sum rule, apply the differential notation separately.
\(\frac{d}{dx}\left(xsin\left(x\right)+32\right)=\frac{d}{dx}\left(xsin\left(x\right)\right)+\frac{d}{dx}\left(32\right)\)
Step 3: Now differentiate the above term.
\(\frac{d}{dx}\left(xsin\left(x\right)+32\right)=sin\left(x\right)\frac{d}{dx}\left(x\right)+x\frac{d}{dx}\left(sin\left(x\right)\right)+\frac{d}{dx}\left(32\right)\)
\(\frac{d}{dx}\left(xsin\left(x\right)+32\right)=sin\left(x\right)\left(1\right)+x\left(cos\left(x\right)\right)+\left(0\right)\)
\(\frac{d}{dx}\left(xsin\left(x\right)+32\right)=sin\left(x\right)+xcos\left(x\right)\)
Step 4: Now apply the differential again on the above first derivative function to get the second order differential.
\(\frac{d}{dx}\left[\frac{d}{dx}\left(xsin\left(x\right)+32\right)\right]=\frac{d}{dx}\left[sin\left(x\right)+xcos\left(x\right)\right]\)
\(\frac{d^2}{dx^2}\left(xsin\left(x\right)+32\right)=\frac{d}{dx}\left[sin\left(x\right)+xcos\left(x\right)\right]\)
Step 5: Apply the sum and product rules and differentiate the function.
\(\frac{d^2}{dx^2}\left(xsin\left(x\right)+32\right)=\frac{d}{dx}\left(sin\left(x\right)\right)+\frac{d}{dx}\left(xcos\left(x\right)\right)\)
\(\frac{d^2}{dx^2}\left(xsin\left(x\right)+32\right)=\frac{d}{dx}\left(sin\left(x\right)\right)+cos\left(x\right)\frac{d}{dx}\left(x\right)+x\frac{d}{dx}\left(cos\left(x\right)\right)\)
\(\frac{d^2}{dx^2}\left(xsin\left(x\right)+32\right)=\left(cos\left(x\right)\right)+cos\left(x\right)\left(1\right)+x\left(-sin\left(x\right)\right)\)
\(\frac{d^2}{dx^2}\left(xsin\left(x\right)+32\right)=cos\left(x\right)+cos\left(x\right)-xsin\left(x\right)\)
\(\frac{d^2}{dx^2}\left(xsin\left(x\right)+32\right)=2cos\left(x\right)-xsin\left(x\right)\)
Table of some 2nd derivatives of functions
Following are a few examples of the second differential solved by our second differential calculator.
| Second derivative of | Output |
| sinx | -sinx |
| e^x | e^x |
| sqrt(x) | \(\frac{-1}{\:4x^{\frac{3}{2}}}\) |
| 1/x | \(\frac{2}{\:x^3}\) |
| a^x | \(a^xlog^2\left(a\right)\) |
Resources
What is Second derivative? | Wikipedia.
