About Second Derivative Calculator
The second derivative is the derivative of the derivative. This page opens with f(x) = x^4 + 3x^2, whose first derivative is 4x^3 + 6x and whose second derivative is 12x^2 + 6. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The second derivative tells you how the slope is changing. Where it is positive the curve is concave up and bends like a bowl, and where it is negative the curve is concave down. A point where the sign changes is an inflection point.
Here 12x^2 + 6 is positive for every x, so x^4 + 3x^2 is concave up everywhere. Use the order buttons to go up to the fifth derivative, or enter a point to see the value of the second derivative there.
How to use Second Derivative Calculator
- Type the function in the Function box, for example x^2*sin(x), or tap keys on the key pad. The preview underneath shows how it was read.
- Choose the order: 1st for the ordinary derivative, up to 5th for higher derivatives.
- To differentiate with respect to another letter, change "With respect to". Other letters are then treated as constants, which gives a partial derivative.
- For an equation such as x^2 + y^2 = 25, switch to Implicit and type both sides with an equals sign.
- Enter a point such as 2 or pi/4 to see the value, the slope and the tangent line there, drawn on the graph.
- Read the steps under the result to see which rule was used at each stage, and copy the answer with the copy button.
Features
- First to fifth derivatives of a function of one variable.
- Partial derivatives: choose any variable and the others are held constant.
- Implicit differentiation of an equation in x and y, solved for dy/dx.
- Value, slope and tangent line equation at a point you choose, with the tangent drawn on the graph.
- Step-by-step working that names the power, product, quotient and chain rules and the standard derivative used.
- A live preview and a math key pad, so input is easy on a phone.
- A zoomable, draggable graph of the function and its derivative that you can save as a PNG.
- Clear errors: input the calculator cannot read is flagged with the position of the problem.
Worked examples
f(x) = x^4 + 3x^2, second derivative gives 12*x^2 + 6
First 4x^3 + 6x, then 12x^2 + 6.
Privacy: how it works
- The derivative is worked out by code running in your browser tab. Your functions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- Use * between factors when you can: 2*x is always read as a product.
- To check a result, integrate it with the integral calculator and see if you get the original function back, up to a constant.
- A tangent line is the best straight-line approximation to the function near the point, which is why it is shown next to the graph.
- For the nth derivative of a function that repeats, such as sin(x), pick the order and watch the pattern: sin, cos, -sin, -cos.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your functions are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your functions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is the second derivative of x^4 + 3x^2?
The first derivative is 4x^3 + 6x. Differentiating again gives 12x^2 + 6.
What does the second derivative tell me about a graph?
Its sign gives the concavity. Positive means concave up, negative means concave down, and a sign change marks an inflection point. At a critical point it also tells you if it is a minimum or a maximum.
How do I find the third derivative?
Choose 3rd in the order buttons. The calculator differentiates three times and shows each round, so you can read the second derivative on the way.
More Derivative Calculator pages
- Derivative CalculatorType a function, get its derivative with every rule named, a tangent line at any point and a graph.
- Derivative of ln(x)Step-by-step derivative of ln(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of sin(x)Step-by-step derivative of sin(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of cos(x)Step-by-step derivative of cos(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of tan(x)Step-by-step derivative of tan(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of sec(x)Step-by-step derivative of sec(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of e^xStep-by-step derivative of e^x, with the rule named at each line and a graph of the function and its slope.
- Derivative of x^2Step-by-step derivative of x^2, with the rule named at each line and a graph of the function and its slope.
- Derivative of sqrt(x)Step-by-step derivative of sqrt(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of 1/xStep-by-step derivative of 1/x, with the rule named at each line and a graph of the function and its slope.
- Derivative of arctan(x)Step-by-step derivative of arctan(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of x*ln(x)Step-by-step derivative of x*ln(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of e^(2x)Step-by-step derivative of e^(2x), with the rule named at each line and a graph of the function and its slope.