Derivative Calculator

How the Derivative Calculator works

A derivative measures how fast a function changes. For f(x) = x², the derivative 2x says that at x = 3 the curve is climbing 6 units up for every unit across. Where the derivative is 0 the curve is flat for an instant, which is how maximum and minimum points are found.

How the derivative calculator works

The calculator reads your function into a tree of operations, then applies one differentiation rule per node and tidies the result. The power rule formula is d/dx xⁿ = n·xⁿ⁻¹. Products use (uv)′ = u′v + uv′, quotients use (u/v)′ = (u′v − uv′)/v², and a function inside another uses the chain rule, multiplying by the derivative of the inside. The tangent line at x = a is y = f′(a)(x − a) + f(a).

First example. f(x) = x³ − 3x gives f′(x) = 3x² − 3. At x = 2 the slope is 9 and f(2) = 2, so the tangent line is y = 9x − 16.

Second example. f(x) = x² sin(x) needs the product rule: f′(x) = 2x·sin(x) + x²·cos(x).

How many turning points does x³ − 3x have? Set 3x² − 3 = 0 and x = 1 or −1, so two: a maximum at (−1, 2) and a minimum at (1, −2).

Values of f(x) = x^3 - 3x and f′(x) from x = −3 to 3

Tips

• Type it as you would write it. 3x^2, x sin(x), e^(2x), sqrt(x) and ln(x) all work.
• log is base 10, ln is natural. e on its own is Euler's number, about 2.71828.
• Check on the graph. Plot the function in the grapher and the derivative's sign shows where it rises and falls.

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