Derivative Calculator
Differentiate any function up to the 4th order, see the rule used and the value at a point
Runs entirely in your browser. Nothing you type or upload is sent to us or stored.
Results are for general guidance. Please check anything important before relying on it. Terms of use
How to Use
- Type your function in the f(x) = box, for example x^2 sin(x) or (x^2 + 1)/(x − 1). The buttons below it insert x², √, sin, cos, tan, ln, log, eˣ, π and a fraction for you.
- Leave the variable as x, or change it (to t, for instance). Choose the order: f′ for the first derivative, up to f⁗ for the fourth.
- If you want the slope at a particular point, type it in Evaluate at (2, −1.5 or pi/4 all work).
- The derivative appears as you type, with the rule that applies (power, product, quotient, chain, trigonometric). Use copy answer for the neat form or copy as text for calculator syntax. For higher orders, each derivative is listed below.
Find derivatives and see which rule applies
This derivative calculator differentiates a function of x and writes the answer the way you would in a notebook: 2x·cos(x²), not 2 * x * cos(x ^ 2). It also names the rule needed for the outermost part of the expression, so you know whether a question is testing the product rule, the quotient rule or the chain rule. You can go up to the fourth derivative and find the value of the derivative at a point, which is the slope of the curve there.
Differentiation is introduced in Class 11 (Limits and Derivatives) and developed in Class 12 (Continuity and Differentiability, and Application of Derivatives). It is also a large part of the calculus section of JEE Main.
The rules at a glance
| Rule | Formula | Example |
|---|---|---|
| Power | d/dx xⁿ = n·xⁿ⁻¹ | d/dx x⁵ = 5x⁴ |
| Product | (uv)′ = u′v + uv′ | d/dx x·sin x = sin x + x cos x |
| Quotient | (u/v)′ = (u′v − uv′)/v² | d/dx sin x / x = (x cos x − sin x)/x² |
| Chain | d/dx f(g(x)) = f′(g(x))·g′(x) | d/dx sin(x²) = 2x cos(x²) |
| Exponential | d/dx eˣ = eˣ, d/dx aˣ = aˣ ln a | d/dx 2ˣ = 2ˣ ln 2 |
| Logarithm | d/dx ln x = 1/x | d/dx log₁₀ x = 1/(x ln 10) |
The trigonometric derivatives you need most often are:
d/dx tan x = sec² x d/dx cot x = −cosec² x
d/dx sec x = sec x tan x d/dx cosec x = −cosec x cot x
Worked example: the quotient rule
Find the derivative of f(x) = (x² + 1)/(x − 1). Take u = x² + 1 and v = x − 1, so u′ = 2x and v′ = 1.
= (2x(x − 1) − (x² + 1)·1)/(x − 1)²
= (2x² − 2x − x² − 1)/(x − 1)²
= (x² − 2x − 1)/(x − 1)²
At x = 3 this gives (9 − 6 − 1)/4 = 0.5, so the curve rises by about half a unit for each unit step in x there. At x = 1 the derivative is undefined, because the function itself is undefined there.
Worked example: the chain rule
For f(x) = (3x + 2)⁵, the outer function is u⁵ and the inner one is u = 3x + 2. Differentiate the outer part and multiply by the derivative of the inner part: f′(x) = 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴. Differentiating again gives f″(x) = 180(3x + 2)³.
How to type functions
- Powers: x^3, or x³. Roots: sqrt(x) or √x. Absolute value: abs(x) or |x|.
- Multiplication can be implied: 2x, 3sin(x) and x sin(x) all work. Write x*sin(x) rather than xsin(x).
- ln is the natural logarithm (base e) and log is base 10, as on a scientific calculator and in most Indian textbooks.
- sin^2 x is read as (sin x)². Angles are in radians, as they must be in calculus: d/dx sin x = cos x is only true in radians.
Good to know
- Answers are simplified, but a calculator's "simplest form" can differ from your book's. Both forms are equal; check by evaluating them at the same point.
- Letters other than the chosen variable are treated as constants: with the variable set to t, d/dt (a t² + 5t) = 2a·t + 5.
- The value at a point is rounded to about 10 significant digits.