Derivative Calculator
Symbolic first and second derivatives, slope and tangent line.
About this tool
Type a function of x and a point. The tool shows the simplified function, its first and second derivatives (each with a copy button), and at your point the value, slope, concavity and the equation of the tangent line.
Differentiation rules
The derivative f′(x) gives the instantaneous rate of change, or the slope of the graph, at each x. This calculator works symbolically using the standard rules: the power rule (d/dx xⁿ = nxⁿ⁻¹), product rule (uv)′ = u′v + uv′, quotient rule, and chain rule for compositions like sin(x²), along with the derivatives of sin, cos, tan, their inverses, sinh, cosh, tanh, eˣ, ln, log, square and cube roots and powers like xˣ. It then simplifies, folding constants and removing zeros and ones. For x³ − 2x + sin x it gives 3x² − 2 + cos x, and the second derivative 6x − sin x. At x = 1, the slope is 1 + cos 1 ≈ 1.54, and a positive second derivative means the curve is concave up there. The tangent line is y = f′(a)(x − a) + f(a). Behind the scenes, the derivative engine was checked against numerical differentiation on thousands of random expressions. Honest limits: the simplifier is light, so answers may look longer than a textbook's while being equal; abs(x) differentiates to sign(x), which is undefined at 0. The integral calculator and the slope calculator are related.
Frequently asked questions
What is the derivative of x²?
2x, by the power rule.
What is the derivative of sin x?
cos x.
What does the second derivative tell you?
How the slope changes: positive means the curve is concave up, negative concave down.
How do I find the tangent line at a point?
Evaluate f(a) and f′(a); the tangent is y = f′(a)(x − a) + f(a).
Is my data sent anywhere?
No. Everything is calculated in your browser, and nothing is stored or sent.