Derivative calculator, with every step shown.

A derivative calculator finds the derivative of a function: a new function that tells you the slope of the original at every point. This one shows the working, naming each rule as it is used, so you can follow it and check your own homework. It handles sums, products, quotients, powers, roots, exponentials, logs, trig and inverse trig functions, hyperbolic functions and absolute value, up to the fifth derivative, and it can give the exact value of the derivative at a point.

How to use the derivative calculator

1. Type your function in Function, for example x^3 sin(x). Multiplication can be left out, so 2x and x sin x both work. Use the symbol buttons to add a power, a root, pi or a function.
2. Check the line that starts "Read as". It shows the function the way the calculator understood it. If a bracket is in the wrong place, fix it here.
3. Leave Variable on Auto, or choose the letter you are differentiating with respect to. Other letters are treated as constants.
4. Pick the Order: 1st for the slope, 2nd for how the slope changes, and so on.
5. To get the slope at one place, type a number or an expression like pi/4 in Value at a point.
6. Read the answer, read the steps below it, and use Copy answer or Copy link. The link opens this page with your function filled in.

The rules, with one example each

The steps use these rules. They are the same ones taught in any calculus course.

• Power rule. The derivative of x^n is n x^(n-1). So the derivative of x^4 is 4x^3, and the derivative of the square root of x is 1/(2 sqrt(x)).
• Product rule. For f times g, the derivative is f' g + f g'. For x^3 sin(x) that gives 3x^2 sin(x) + x^3 cos(x).
• Quotient rule. For f over g it is (f' g − f g') / g^2. For ln(x)/x, f = ln(x) and g = x, so the top is (1/x)(x) − ln(x)(1) = 1 − ln(x), and the answer is (1 − ln(x)) / x^2.
• Chain rule. For a function of a function, differentiate the outside and multiply by the derivative of the inside. For sin(x^2) the outside gives cos(x^2) and the inside gives 2x, so the answer is 2x cos(x^2).
• Exponential and log. The derivative of e^x is e^x, of a^x is a^x ln(a), and of ln(x) is 1/x.

Standard derivatives the calculator knows

A longer worked example

Take e^(2x) cos(x). It is a product, so use the product rule with f = e^(2x) and g = cos(x). The derivative of f needs the chain rule: the outside e^u gives e^(2x) and the inside 2x gives 2, so f' = 2e^(2x). The derivative of g is -sin(x). Put it together: f' g + f g' = 2e^(2x) cos(x) − e^(2x) sin(x). Type it in and the calculator shows the same pieces in the steps.

Derivative at a point

Type (x^2+1)/(x-1) and put 3 in Value at a point. The derivative is (x^2 − 2x − 1)/(x − 1)^2. At x = 3 the top is 9 − 6 − 1 = 2 and the bottom is 2^2 = 4, so the slope is 1/2. The calculator shows 1/2 exactly, then the decimal 0.5.

Higher derivatives

Choose a higher Order to differentiate again. For x^4 the first derivative is 4x^3, the second is 12x^2, and the third is 24x. Each order is worked from the one before, and the steps show all of them.

What it cannot do

It differentiates with respect to one letter at a time (a partial derivative, with the other letters as constants). It does not do implicit differentiation, limits or functions defined in pieces. If your function has a rule it cannot read, the box tells you what to change. Absolute value is differentiated away from 0 only.

Common problems

• A result that looks too long is often correct but not factored. The calculator combines fractions and expands where that is shorter, but a hand answer can look different and still be equal. Compare values at a point to be sure.
• Check that "Read as" matches what you meant. Writing 1/2x reads as (1/2) times x. For 1 over 2x, write 1/(2x).
• log(x) is the base 10 logarithm here. For the natural log, use ln(x).

To draw the function on its own, use the graphing calculator. For the slope between two points, try the slope calculator, and for general sums the scientific calculator.