Generalized Power Rule Calculator

What is the Generalized Power Rule?

The Generalized Power Rule is a fundamental rule in calculus used to find the derivative of a power function. For any real number n, the derivative of f(x) = xⁿ is given by:

f'(x) = nx^n-1

This rule extends the basic power rule to non-integer exponents, making it applicable to a wider range of functions.

How to Use This Calculator

1. Enter the function base (typically 'x') in the "Function Base" field
2. Enter the power (n) in the "Power" field. This can be any real number (integer, fraction, or decimal)
3. Click the "Calculate" button to compute the derivative
4. The result will be displayed in the "Derivative Result" area

Examples

Example 1: For f(x) = x³, enter x as the base and 3 as the power. The derivative will be 3x².

Example 2: For f(x) = √x (which is x^1/2), enter x as the base and 0.5 as the power. The derivative will be 0.5x^-0.5 or (1/2√x).

Example 3: For f(x) = 1/x (which is x^-1), enter x as the base and -1 as the power. The derivative will be -1x^-2 or -1/x².

Mathematical Background

The generalized power rule is derived from the limit definition of a derivative:

f'(x) = lim_h→0 [(x+h)ⁿ - xⁿ]/h

When simplified, this limit produces the formula f'(x) = nx^n-1 for all real numbers n.