📐 Derivative Calculator
The derivative is where calculus starts: it measures how fast a function changes, and it is the slope of the tangent line at any point. This calculator differentiates your function symbolically — it returns the exact formula for f′(x), applying the power, product, quotient and chain rules the way a CAS does.
Type any function of x — polynomials, trig, exponentials, logs, roots, quotients, even x^x — and optionally ask for the value of the derivative at a specific point.
How to use this calculator
Write the function in terms of x. Implied multiplication like 2x and 3sin(x) works, and angles are in radians.
- Type your function f(x) — for example x^3 + 2*sin(x), ln(x)/x or (x^2+1)^3.
- Optionally enter a point in the second box to also get f and f′ evaluated there — f′ at a point is the tangent slope.
- Press Calculate and read the exact derivative f′(x), simplified.
- Check your homework by evaluating: your by-hand answer and this one must give the same number at any x.
Derivative Worked Example
Differentiate f(x) = x³ + 2·sin(x). The power rule turns x³ into 3x², and the derivative of sin(x) is cos(x), so the constant multiple gives 2·cos(x). Together: f′(x) = 3x² + 2cos(x). At x = 1 the tangent slope is f′(1) = 3 + 2·cos(1) = 3 + 2(0.5403) = 4.0806 — type 1 in the point box and the calculator confirms both f(1) and f′(1).
Frequently asked questions
How does this derivative calculator work?
It differentiates symbolically — the same job a CAS (computer algebra system) like a TI-89 does. Your expression is parsed into a tree, the power, product, quotient and chain rules are applied step by step, and the result is simplified before printing. That is why you get 3x² + 2cos(x) as an exact formula, not a decimal approximation.
Which functions can I differentiate?
Polynomials of any size, sin, cos, tan and their inverses, sinh, cosh, tanh, exp, ln, log (base 10), sqrt and cbrt — combined with products, quotients, powers and nesting, including general powers like x^x. Angles are in radians. The one common exception is abs(x), whose derivative is undefined at 0.
What does 'evaluate at a point' give me?
The value of f′(x) at your chosen x — geometrically, the slope of the tangent line to the curve at that point. Enter it in the optional second box and the calculator shows both f(x) and f′(x) there.
My derivative looks different from my textbook's. Is it wrong?
Usually it is the same answer in a different dress: 1/cos(x)² is sec²(x), and -sin(x)/x² forms can be rearranged several ways. The reliable check is to evaluate both versions at one x value (try the point box) — if they agree numerically, they are the same derivative.