What does this derivative calculator find?
A derivative measures a function’s instantaneous rate of change. This calculator differentiates a real, single-variable expression and shows the calculus rules applied to your input. Choose a first, second, third or fourth derivative. You can optionally evaluate the resulting derivative at a specific point to estimate a tangent slope or a higher-order rate.
For example, if f(x) = x^3 - 2x, then f′(x) = 3x^2 - 2. At x = 2 the slope is 10. The Graphing Calculator can help you explore the original curve, while the Integral Calculator performs the reverse operation of finding an antiderivative.
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For independent practice, optional calculus workbooks (paid link) can provide exercises to work through before checking your result. No purchase is needed to use this calculator.
How to calculate a derivative with steps
Enter a function, select the variable and derivative order, then choose Calculate. Leave the evaluation point blank for a symbolic expression. Enter a point only when you also want a rounded numerical value. Changing any input clears the previous result; Reset returns the form to its initial state. Copy, Download and Print include the calculation steps.
For a composite function such as sin(x^2), keep the entire inner expression in parentheses. For a quotient, write (x^2+1)/(x-1). The Simplifying Expressions Calculator can help organize algebra before differentiating. If you keep hand calculations in a graph-paper notebook (paid link), label the original expression, the rule and the derivative on separate lines.
Expression syntax and angle conventions
Use your chosen variable (x, y or t), numbers, pi, e, parentheses and +, -, *, / or ^. Multiplication may be explicit, as in 2*x, or implicit, as in 2x or (x+1)(x-1). Write sin(x), cos(x), tan(x), exp(x), ln(x), sqrt(x), abs(x), asin(x), acos(x) or atan(x). Functions need parentheses. log(x) means the natural logarithm, just like ln(x). All trigonometric angles are in radians; inverse trigonometric answers are also in radians.
Powers associate to the right: x^2^3 means x^(2^3). A leading minus has lower precedence than a power, so -x^2 means -(x^2); use (-x)^2 to square the negative input. Only one selected variable is allowed. Bounds and evaluation points may be constant expressions such as pi/2, sqrt(2) or 1/3.
Power, product, quotient and chain rules
The constant rule gives d(c)/dx = 0. The power rule gives d(x^n)/dx = n*x^(n-1). Sums and differences are differentiated term by term. The product rule gives (u*v)′ = u′*v + u*v′, and the quotient rule gives (u/v)′ = (u′*v - u*v′)/v^2 wherever the denominator is nonzero.
The chain rule multiplies an outer derivative by the inner derivative. Thus d(sin(x^2))/dx = cos(x^2)*2x. The derivative of exp(u) is exp(u)*u′; the derivative of ln(u) is u′/u on its real domain. Constant-base exponentials and variable powers are supported; logarithmic differentiation for variable powers requires a positive base. Higher derivatives repeat the same process.
Rational coefficients are manipulated as exact fractions within the engine’s size limits. The expression may remain factored or contain extra parentheses; this does not change the underlying calculus rule. The Fraction Calculator is useful for separately reviewing coefficient arithmetic.
Worked derivative examples
Polynomial: For f(x) = x^3 - 2x, differentiate each term to get f′(x) = 3x^2 - 2. Substitution at x = 2 gives 3*4 - 2 = 10. Differentiating again gives f″(x) = 6x, so the second derivative at x = 2 is 12.
Chain rule: For f(x) = sin(x^2), the outer sine becomes cosine and the inner derivative is 2x. Therefore f′(x) = cos(x^2)*2x. At x = 0 the derivative is 0.
Quotient: For f(x) = (x^2+1)/(x-1), the numerator of the derivative is 2x*(x-1) - (x^2+1), and its denominator is (x-1)^2. The original function and derivative are both undefined at x = 1. Optional scientific calculators (paid link) can help check numerical substitution, depending on the model’s supported functions.
What the result and evaluation point mean
The main expression is the selected derivative, not the original function value. The result facts identify the derivative order. When a point is supplied, the facts also show the rounded derivative value and the original function value at that point. A positive first derivative indicates local increase, while a negative one indicates local decrease where the derivative exists.
A zero derivative at a point is not by itself proof of a maximum or minimum. Likewise, a decimal approximation is not an exact symbolic proof. Optional graphing calculators (paid link) can help compare a curve and a slope numerically; match the function, angle mode and viewing window before comparing outputs.
Supported functions, real domains and limits
This version supports the listed elementary functions and one variable, with derivative orders 1 through 4. It does not solve implicit equations, partial derivatives or piecewise functions. abs(u) uses u/abs(u) times the inner derivative away from inner zeros. It may conservatively reject evaluation at such a zero even when a surrounding expression has a removable nondifferentiability. Algebraic simplification never authorizes filling a hole in the original function.
A symbolic expression applies only where the original function and required derivatives exist; this tool does not produce a complete domain solution. ln requires a positive argument, sqrt requires a nonnegative argument for the original function and may have an undefined derivative at zero, and denominators must stay nonzero. Noninteger powers of negative bases are outside the numerical real-power convention. Variable-power derivative formulas require positive bases.
Inputs are limited to 240 characters, 160 tokens and 24 nested levels. Numeric literals use at most 24 digits and decimal exponent magnitude 30. Rational coefficient size, symbolic work and worker runtime are bounded. Numerical points must lie between -1,000,000 and 1,000,000; values are rounded to roughly 11 significant digits. Very complex expressions may need a shorter input or a lower derivative order.
Derivative Calculator FAQs
Sources and methodology
These educational references explain the underlying calculus. This independently implemented browser tool is not affiliated with their publisher. The supported forms, numerical method and limits are described above.