Power rule

The power rule is in mathematics one of the basic rules of differential calculus. It is used to determine the derivative of power functions. It reads:

Is example.

Generalization

The power rule also applies to power functions whose exponent ( exponent ) is not an integer:

Derivation

Case 1: The exponent is a natural number

The derivative of a power function of the position x is the threshold:

According to the binomial theorem this is equal to

Written with so-called binomial coefficients. It follows then the power rule:

Pictorially illustrates a growing 'n - dimensional cube ' in exactly n directions ( along the n coordinate axes ) to '(n -1 )-dimensional cube ' on. A square is growing (or crystallized ) so marginally by 2 sidelines, and a cube grows by 3 squares.

Case 2: Generic exponent

One uses the representation using the exponential function and forwards using the chain rule and the derivation rule for the exponential function:

For the inner discharge to use the factor rule and the rule for deriving the logarithm:

By employing this and writes for again, one obtains

This derivation is only valid for. For the function is differentiable but also at the position, and the rule is also valid at the point. We calculated directly using the differences quotient: