Power Rule Calculator — Derivative of axⁿ
Apply the power rule to find the derivative of f(x) = axⁿ.
Evaluate both the function and its derivative at any point x.
Covers constant, linear, and polynomial terms.
The power rule is the most-used differentiation rule in calculus:
If f(x) = a * x^n, then f’(x) = n * a * x^(n-1)
Bring the exponent down as a multiplier, then reduce the exponent by 1. That is it.
A few cases worth noting: the derivative of a constant (n=0) is always zero – no x, no rate of change. The derivative of a linear term ax (n=1) is just a. The derivative of x^2 is 2x, x^3 is 3x^2, and so on.
The rule extends to negative and fractional exponents without any extra steps. The derivative of x^(-2) = -2x^(-3). The derivative of x^(1/2) (the square root) is (1/2)x^(-1/2) = 1/(2*sqrt(x)).
This calculator also evaluates f(x) and f’(x) at a specific x value. The derivative at a point gives the slope of the tangent line there – the instantaneous rate of change at that exact input. If you need the tangent line equation itself, use the Tangent Line Calculator.
For sums of power terms (a full polynomial), apply the power rule to each term independently and add the results. The derivative of 3x^4 + 2x^2 - 5 is 12x^3 + 4x.