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Mathematics

Derivative Calculator

Type a function, get its derivative with every rule named, a tangent line at any point and a graph.

Kind of derivative
x2sin⁡xx^{2}\sin xx2sinx
Order of the derivative
Use y or t for a partial derivative.
Adds the slope and the tangent line.

Derivative

ddx[x2sin⁡x]=x2cos⁡x+2xsin⁡x\frac{d}{d x}\left[x^{2}\sin x\right]=x^{2}\cos x+2x\sin xdxd​[x2sinx]=x2cosx+2xsinx

At x = 1

f(1) = 0.84147098

f'(1) = 2.2232443

Tangent line

y=2.2232443x−1.3817733y=2.2232443x-1.3817733y=2.2232443x−1.3817733
  • f(x)
  • f'
  • tangent
Table of values
Graph of x^2*sin(x) and its derivative: sampled values across the visible range
xf(x)f'tangent
-523.973-2.4977-12.498
-3.88.8352-16.072-9.8301
-2.6-3.4848-3.112-7.1622
-1.4-1.93153.0924-4.4943
-0.2-0.00794680.11867-1.8264
10.841472.22320.84147
2.23.91310.709043.5094
3.4-2.9541-12.9146.1773
4.6-21.027-11.5158.8452
5.8-15.62924.39911.513
732.19246.13914.181

Step-by-step working

  1. 1Product rule
    ddx[x2sin⁡x]\frac{d}{dx}\left[x^{2}\sin x\right]dxd​[x2sinx]
    ↓
    ddx[x2]⋅sin⁡x+x2⋅ddx[sin⁡x]\frac{d}{dx}\left[x^{2}\right]\cdot \sin x+x^{2}\cdot \frac{d}{dx}\left[\sin x\right]dxd​[x2]⋅sinx+x2⋅dxd​[sinx]
  2. 2Power rule
    ddx[x2]\frac{d}{dx}\left[x^{2}\right]dxd​[x2]
    ↓
    2x2x2x
  3. 3Derivative of \sin x
    ddx[sin⁡x]\frac{d}{dx}\left[\sin x\right]dxd​[sinx]
    ↓
    cos⁡x\cos xcosx
  4. 4Simplify
    2x1sin⁡x+x2cos⁡x2x^{1}\sin x+x^{2}\cos x2x1sinx+x2cosx
    ↓
    x2cos⁡x+2xsin⁡xx^{2}\cos x+2x\sin xx2cosx+2xsinx
  1. Tools
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  3. Derivative Calculator

About Derivative Calculator

This derivative calculator is free, needs no sign-up and runs in your browser, so the functions you type are never uploaded. Enter a function of x and it differentiates it, writes out each rule it used, and plots the function next to its derivative.

It goes beyond a single answer. You can take the first up to the fifth derivative, differentiate with respect to another variable to get a partial derivative, switch to implicit mode to find dy/dx from an equation in x and y, or give a point to read off the slope and the equation of the tangent line there.

The input field shows a live preview in proper notation as you type and has a key pad for powers, roots, logarithms and trigonometric functions, so it works on a phone as well as on a keyboard.

How to use Derivative Calculator

  1. 1Type 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.
  2. 2Choose the order: 1st for the ordinary derivative, up to 5th for higher derivatives.
  3. 3To differentiate with respect to another letter, change "With respect to". Other letters are then treated as constants, which gives a partial derivative.
  4. 4For an equation such as x^2 + y^2 = 25, switch to Implicit and type both sides with an equals sign.
  5. 5Enter a point such as 2 or pi/4 to see the value, the slope and the tangent line there, drawn on the graph.
  6. 6Read 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.

The rules the calculator uses

Every derivative here is built from a short list of rules. The power, sum and constant multiple rules handle polynomials, the product and quotient rules combine functions, and the chain rule handles a function inside another function.

The derivatives of the standard functions are applied directly: sine, cosine, tangent, the exponential, the logarithm and the inverse trigonometric functions.

(x^n)' = n*x^(n-1)
(u*v)' = u'*v + u*v'
(u/v)' = (u'*v - u*v')/v^2
(f(g(x)))' = f'(g(x))*g'(x)
(e^x)' = e^x
(ln x)' = 1/x

Worked examples

  • d/dx x^3*sin(x) → gives x^3*cos(x) + 3*x^2*sin(x)

    Product rule: (x^3)' sin(x) + x^3 (sin(x))'.

  • Second derivative of e^(2x) → gives 4*e^(2*x)

    Each differentiation brings down a factor of 2.

  • Partial derivative of x^2*y + sin(x*y) with respect to x → gives 2*x*y + y*cos(x*y)

    y is held constant.

  • dy/dx for x^2 + y^2 = 25 → gives -x/y

    Differentiate both sides, treating y as a function of x.

  • Tangent to x^2 at x = 3 → gives y = 6x-9

    The slope is 2x = 6 and the point is (3, 9).

Good to know

What the calculator can and cannot do

It differentiates symbolically for functions built from polynomials, powers and roots, exponentials, logarithms, the trigonometric and inverse trigonometric functions and absolute value. The result is simplified, but the same derivative can be written in more than one correct form, such as 2 sin(x) cos(x) and sin(2x).

Log means base 10 and ln means the natural logarithm. Trigonometric functions use radians. Write a product with a star, as in x^x*(ln(x)+1), when you check an answer by typing it back in.

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.

How do I find the derivative of a function?

Break the function into pieces and apply the rule that matches each piece: the power rule for x^n, the product or quotient rule when two functions are multiplied or divided, and the chain rule when one function sits inside another. The calculator names the rule at each step.

How do I take a second or higher derivative?

Differentiate the result again. Choose 2nd to 5th in Order and the calculator repeats the process and shows each round.

What is a partial derivative?

It is the derivative of a function of several variables with respect to one of them, treating the others as constants. Change "With respect to" to the variable you want.

What is implicit differentiation?

It finds dy/dx when y is not written out as a function of x, such as in a circle. You differentiate both sides, treat y as a function of x so each y term picks up dy/dx, and solve for dy/dx.

How do I get the equation of the tangent line?

Enter a point. The calculator finds the slope f'(a) and the value f(a), then writes y = f'(a)(x - a) + f(a) in simplified form and draws it on the graph.

Does it use radians or degrees?

Radians. The derivative of sin(x) is cos(x) only in radians, so every trigonometric function here is in radians.

Why does my answer look different from my textbook?

A derivative can be written in several equal forms. For example 2 sin(x) cos(x) equals sin(2x), and x^(-1) equals 1/x. The values agree, and you can check by plugging in a number.

Popular Derivative Calculator pages

  • 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.
  • Derivative of sin^2(x)Step-by-step derivative of sin^2(x), with the rule named at each line and a graph of the function and its slope.
  • Derivative of ln(ln(x))Step-by-step derivative of ln(ln(x)), with the rule named at each line and a graph of the function and its slope.
  • Derivative of x^xStep-by-step derivative of x^x, with the rule named at each line and a graph of the function and its slope.
  • Second Derivative CalculatorDifferentiate twice, see both rounds of working, and read off where the curve bends up or down.
  • Partial Derivative CalculatorDifferentiate with respect to one variable while the others stay constant.
  • Implicit Differentiation CalculatorEnter an equation in x and y and get dy/dx without solving for y first.

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