Power Rule - Derivatives

The power rule can be applied to expressions containing a variable with a constant exponent, such as x², x^-4, x^-1/3, and x^2/5. It also applies to variables having complex number exponents, such as xⁱ and x^{(2 + 3i)}.

The power rule does not apply to expressions in which a constant has a variable exponent, such as e^x or 10^x. It does not apply to variables having variable exponents as well, such as x^x.

The power rule can be represented by the following formula:

`d/(dx) x^n = nx^(n-1)`

…where x is a variable and n is a number. In words, the power rule says that if a variable has an exponent of a number, say n, then its derivative is given by n times the variable with an exponent of one less than n.

The following procedure sums up how power rule is applied,

• Take the exponent on the variable, and place it before the variable
• Reduce the exponent on the variable by 1 and let this become the new exponent on the variable

The exponent on the variable can be a negative or positive number, zero, a fraction, a decimal number or a complex number.

For example, consider the following function,

`f(x) = x^5`

To differentiate it, the power rule is to be applied, because there is a variable with a constant exponent. The variable is x, and exponent is 5.

• Take the exponent 5 and place it before the variable
• Take the exponent 5, reduce it by 1 to get 4, and let this become the new exponent on the variable

The derivative is,

`f'(x) = 5x^4`