Derivative Calculator Online
This derivative calculator finds the exact derivative of a function as you type and shows it typeset like a textbook, next to the function you entered. Add a point and it also gives the value of the derivative there. It returns the final derivative only, with no step-by-step working, and everything runs in your browser with no sign-up.
How the derivative calculator works
Type your function into the Function field, for example 3x^2 + 2x. After a
short pause the result card shows two lines: Function, f(x) as the calculator read it, and
Derivative, f'(x) in simplified form. For 3x^2 + 2x that is
6x + 2.
While an expression is half-typed, the card keeps the last good result and dims it after a second, with the
note "Keeping the last result until the expression is complete." Finish the expression and the result
updates. Answers are exact fractions and symbols, such as 1/(3x^(2/3)) for the derivative of
cbrt(x). If you type a decimal such as 0.5x^2, the answer keeps decimals.
How to type a function
Write the function the way you would in a spreadsheet or a graphing calculator. The hint under the field
sums it up: powers as x^2, products as 3x or 2*x, and functions like
sin(x), sqrt(x), e^x and ln(x). Put brackets after a
function name, as in sin(2x). A lone letter or number such as sin x is
understood too, but sin 2x is not.
| You type | It means |
|---|---|
x^2, x^(1/3) | Powers, including fractional and negative powers |
3x, 2*x, x(x+1), 2sin(x) | Multiplication, written out or implied |
ax | a times x, where a is treated as a constant |
pi or π, e | The constants pi and Euler's number |
ln(x), log(x) | Natural logarithm |
log(x, 2), log10(x), log2(x) | Logarithm to another base |
sqrt(x), cbrt(x), nthRoot(x, 4) | Square, cube and nth roots |
abs(x), exp(x) | Absolute value, e to the power x |
Trigonometric functions: sin, cos, tan, sec,
csc, cot, their inverses asin, acos, atan
(or arcsin and so on), and the hyperbolic functions sinh, cosh,
tanh, sech, csch, coth with inverses such as
asinh. Decimals use a point.
Find the derivative at a point
Open Options and type a value under Evaluate the derivative at. A card then
shows f'(value) = the number, to 10 significant digits. The value can be a number or a constant expression
such as pi/2. If the derivative does not exist there, the card says "undefined at this point".
Derivative: 6x + 2
Evaluate the derivative at: 2
f'(2) = 14
Differentiate with respect to another variable
The default variable is x. To differentiate with respect to another letter, such as t, change
Differentiate with respect to under Options. Every other letter in the
function is treated as a constant, and the calculator tells you which ones. So a sin(t) with
respect to t gives a cos(t).
Derivative examples
ln(x^2 + 1) gives 2x/(x^2 + 1)
(x^2 + 1)^3 gives 6x(x^2 + 1)^2
cbrt(x) gives 1/(3x^(2/3))
2^x gives 2^x ln(2)
Results from the product, quotient and chain rules are combined into one expression, but not always fully factored, so your textbook answer may look slightly different yet be equal. The results were checked against a numerical difference quotient on about 90 test functions.
What this derivative calculator does not do
It finds the first derivative of a function of one variable and shows the result only. It does not give steps, and it does not do implicit differentiation, partial or higher-order derivatives, or integrals. It cannot differentiate erf, floor, ceil, round, sign, gamma, factorials or piecewise functions, and says "Can't differentiate erf yet." (or the name of the function) when you try.
Frequently asked questions
How do I find the derivative at a point with a calculator?
Type the function, open Options and enter the point under Evaluate the derivative at. A card then shows the derivative's value at that point, to 10 significant digits. The point can be a plain number such as 2 or a constant expression such as pi/2. For 3x^2 + 2x at 2 the card shows 14.
What is the derivative of sin(x)/x?
By the quotient rule, the derivative of sin(x)/x is (x cos(x) - sin(x))/x^2. Type sin(x)/x into the Function field and the calculator returns exactly that, already combined into a single fraction.
What is the derivative of e^x and ln x?
The derivative of e^x is e^x, and the derivative of ln x is 1/x. In the Function field write e^x and ln(x); the calculator treats both ln and log as the natural logarithm, and log(x, 2) as a logarithm to base 2.
How do I take the derivative of a cube root or square root?
Type cbrt(x) for a cube root or sqrt(x) for a square root. The derivative of cbrt(x) comes back as 1/(3x^(2/3)) and the derivative of sqrt(x) as 1/(2 sqrt(x)). You can also write a fractional power, such as x^(1/3).
How do I differentiate with respect to a variable other than x?
Open Options and type the letter under Differentiate with respect to, for example t. Every other letter in the function is then treated as a constant, so a sin(t) gives a cos(t). The variable must be a single letter, and e is reserved for Euler's number.
Does the derivative calculator show steps?
No. It returns the final derivative only, not the working. If you need to see which rule was used, the derivative of a product, quotient or composite function is combined into one expression, so you can check it against the product, quotient or chain rule yourself.
Can it do partial, implicit or second derivatives?
No. It finds the first derivative of a function of one variable. Implicit differentiation, partial and higher-order derivatives, integrals and piecewise functions are not supported, and functions such as erf, floor, ceil, round, sign, gamma and factorials show a message saying they cannot be differentiated yet.
Does the derivative calculator send my function to a server?
No. The expression is parsed and differentiated in your browser, so nothing you type is sent anywhere, and there is no account or sign-up.
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