Derivative Calculator
The power rule is the most-used rule in differential calculus: the derivative of a coefficient times x raised to an exponent is the coefficient times the exponent, with the exponent reduced by one. This calculator applies that rule to a single power term a*x^n. Enter the coefficient a, the exponent n, and an x value, and it returns the derivative's new coefficient, its new exponent, the symbolic derivative, and the slope of the curve evaluated at your chosen x.
Power rule formula
f(x) = a * x^n
f'(x) = a * n * x^(n - 1)
Derivative coefficient = a * n
Derivative exponent = n - 1
Slope at x = a * n * x^(n - 1)
The derivative multiplies the coefficient by the exponent and lowers the exponent by one. Evaluating at a value of x gives the slope of the tangent line there. The constant case n = 0 yields a derivative of zero.
Differentiation context
- The derivative is the instantaneous rate of change, the slope of the tangent line.
- The power rule holds for any real exponent, including negative and fractional values.
- The derivative of a constant (n = 0) is always zero.
- If n - 1 is negative, the slope is undefined at x = 0.
- For fractional exponents, x must be positive to give a real result.
Derivative: frequently asked questions
What is the power rule for derivatives?
The power rule states that the derivative of a*x^n with respect to x is a*n*x^(n-1). It is one of the foundational rules of differential calculus and applies to any real exponent n. This calculator applies the power rule to a single power term and evaluates the result at a chosen x.
What is a derivative?
A derivative measures the instantaneous rate of change of a function: the slope of the tangent line to its graph at a point. For a position-versus-time graph the derivative is velocity; for a cost curve it is marginal cost. This tool gives both the symbolic derivative coefficient and exponent and its value at a point.
Does this handle a constant term?
Set the exponent n to 0. Then a*x^0 equals the constant a, and the power rule gives a derivative of a*0*x^(-1) = 0. The derivative of any constant is zero, which the calculator confirms.
Can the exponent be negative or fractional?
Yes. The power rule a*n*x^(n-1) holds for any real exponent. A negative exponent (such as 1/x, where n = -1) or a fractional exponent (such as a square root, where n = 0.5) both work, as long as x is in the domain of the original function.
Why is the derivative undefined at some x values?
If the resulting exponent n-1 is negative, the derivative involves division by a power of x and is undefined at x = 0. For fractional exponents, x must be positive for a real value. The calculator returns n/a when the evaluation is not a finite real number.
Official sources
- National Institute of Standards and Technology: Digital Library of Mathematical Functions.
- MIT OpenCourseWare: Single Variable Calculus.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.