voimasääntöön

Voimasääntöön is a Finnish term that translates to "power rule" or "rule of powers" in English. It refers to a fundamental principle in calculus used to differentiate functions of the form x^n, where x is a variable and n is a constant exponent. The rule states that the derivative of x^n with respect to x is n times x raised to the power of n-1. Mathematically, this is expressed as d/dx(x^n) = nx^(n-1).

This rule is a cornerstone of differential calculus and simplifies the process of finding derivatives for many algebraic functions. It applies to positive, negative, and fractional exponents. For example, the derivative of x^2 is 2x^(2-1) = 2x. The derivative of x^3 is 3x^(3-1) = 3x^2. For negative exponents, the derivative of x^-1 is -1x^(-1-1) = -x^-2, which can also be written as -1/x^2. For fractional exponents, such as the square root of x (which is x^(1/2)), the derivative is (1/2)x^((1/2)-1) = (1/2)x^(-1/2), or 1/(2*sqrt(x)).

The voimasääntöön is often one of the first differentiation rules taught in calculus courses due to its broad applicability and foundational nature for understanding more complex differentiation techniques. It can be combined with other rules, such as the constant multiple rule and the sum/difference rule, to differentiate a wide array of polynomial and other algebraic functions.