Derivative Calculator

What does the derivative calculator do?

It differentiates the function you type symbolically, so the result is a formula and not a number. It shows which rule is applied in which order, simplifies the result, evaluates it at a point if you wish and plots the function together with its derivative. Second, third and fourth derivatives are available too.

Rules used

• Sum: (u ± v)′ = u′ ± v′
• Constant multiple: (c·u)′ = c·u′
• Product: (u·v)′ = u′·v + u·v′
• Quotient: (u/v)′ = (u′·v − u·v′) / v²
• Power: (uⁿ)′ = n·uⁿ⁻¹·u′
• Exponential: (eᵘ)′ = eᵘ·u′ and (aᵘ)′ = aᵘ·ln a·u′
• Chain: (f(u))′ = f′(u)·u′

Supported functions: sin, cos, tan, cot, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh, ln, log, log2, sqrt, cbrt, exp and abs.

Example

Type x^2*sin(x). In the product rule take u = x² and v = sin x. Since u′ = 2x and v′ = cos x the derivative is 2x·sin x + x²·cos x. The tool writes the same result as x^2·cos(x) + 2x·sin(x).

How is the result checked?

Every result is compared at several points with the numeric derivative of the function (five point central difference). If the two disagree the tool says so. When you enter a point, the value of the symbolic derivative there is shown next to the numeric derivative.

Limits

Letters other than the variable are treated as constants: the derivative of a*x^2 with respect to x is 2a·x. Angles are in radians. Simplification collects like terms and combines powers, but it does not apply trigonometric identities and does not factor, so a result may look different from the textbook answer while being equal to it. During simplification expressions such as x/x become 1, so the result may appear defined at points where the original function is not. Step functions such as floor, round, mod, max, min and the factorial are not differentiated. Implicit differentiation and systems of partial derivatives are outside the scope of this tool.