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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 x2/5. 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 ex or 10x. It does not apply to variables having variable exponents as well, such as xx.

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`

1 comment:

  1. Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53.

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