Derivative Calculator
Enter any expression and get a full step-by-step solution — free, instantly.
Try an example:
This tool is for educational assistance. Accuracy is not guaranteed; please verify all results independently.
Differentiation Rules Supported
Power Rule
d/dx[xⁿ] = n·xⁿ⁻¹
Sum / Difference Rule
(f ± g)' = f' ± g'
Constant Multiple Rule
(c·f)' = c·f'
Product Rule
(f·g)' = f'g + fg'
Quotient Rule
(f/g)' = (f'g − fg')/g²
Chain Rule
f(g(x))' = f'(g(x))·g'(x)
Exponential (eˣ)
d/dx[eˣ] = eˣ
Natural Log
d/dx[ln x] = 1/x
Trig Functions
sin, cos, tan and their inverses
Square Root
d/dx[√x] = 1/(2√x)
Hyperbolic Functions
sinh, cosh, tanh and their inverses
How to Use
- Type your expression using standard math notation (e.g.
x^2 * sin(x)or2x·sin(x)). - Use
^for exponents,*for multiplication (or write terms together —2xis understood), and/for division. - Trig:
sin,cos,tan,asin,acos,atan. Hyperbolic:sinh,cosh,tanh,asinh,acosh,atanh. Other:ln,exp,sqrt. - Select the variable you want to differentiate with respect to (default: x).
- Press Calculate Derivative or hit Enter.
Common Derivatives
| f(x) | f′(x) |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| eˣ | eˣ |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | −sin(x) |
| tan(x) | sec²(x) |
| arcsin(x) | 1/√(1−x²) |
| arccos(x) | −1/√(1−x²) |
| arctan(x) | 1/(1+x²) |
| √x | 1/(2√x) |
| sinh(x) | cosh(x) |
| cosh(x) | sinh(x) |
| tanh(x) | sech²(x) |
| arcsinh(x) | 1/√(1+x²) |
| arctanh(x) | 1/(1−x²) |
Learn the Rules
The calculator shows you which rule applies at every step — these calculus guides explain the rules themselves, as a learning path written for first-time students:
- Intro to Derivatives — what a derivative measures, and how to read prime and Leibniz notation.
- Basic Rules — the power rule plus constants, constant multiples, and sums: everything you need for polynomials.
- Product & Quotient Rules — what to do when functions are multiplied or divided.
- Derivatives of Transcendental Functions — trig, exponentials, and logarithms, with the logic that makes them stick.
- Mastering the Chain Rule — composite functions, the outside-inside method, and calculus’s most common mistake.
- Applications of Derivatives — velocity, marginal cost, optimization, and other real-world uses.
- 7 Common Derivative Mistakes — a review of the errors graders see most, shown wrong and right side by side.
Frequently Asked Questions
What is a derivative?
A derivative measures how fast a function is changing at any given point — the slope of the curve at that exact spot. If f(x) describes a car’s position over time, its derivative f′(x) describes the car’s speed. Finding derivatives, called differentiation, is one of the two central operations of calculus.
How does the step-by-step solver work?
The calculator parses your expression into a tree, then differentiates it symbolically by applying the standard rules — power rule, product rule, quotient rule, chain rule, and the function-specific rules for trig, logs, and exponentials. Each application of a rule is recorded and displayed as a named step, so the solution reads the way a worked textbook example does.
Is the calculator really free?
Yes — every feature, including full step-by-step solutions and higher-order derivatives, is free with no account, sign-up, or usage limit. The site is supported by advertising.
What functions are supported?
Polynomials, rational functions, square roots, exponentials, natural logarithms, the six trigonometric functions and their inverses, and the hyperbolic functions sinh, cosh, and tanh with their inverses — in any combination, nested to any depth.
Can it compute second and higher-order derivatives?
Yes. Use the Order selector to compute up to the fourth derivative. The calculator shows the step-by-step work for every order along the way, not just the final one.
Is anything I type collected or stored?
No. All computation happens locally in your browser — expressions you enter are never sent to a server, and the site has no accounts or analytics of its own. See the privacy policy for full details.