Final answer:
The power rule for negative exponents is validated using the quotient rule by differentiating a function with a negative exponent and showing that it produces results consistent with the power rule, which concludes the derivative of x to the power of a negative integer is the negative of the integer times x to one less than the negative integer.
Step-by-step explanation:
To prove that the power rule is valid for negative whole number powers using the quotient rule, let us consider the quotient of two functions, where one of them is x raised to a negative integer power. The power rule asserts that when we differentiate xn, where n is any real number, we get n*xn-1. To demonstrate this for negative powers, let's differentiate f(x) = x-n, which we can write as f(x) = 1 / xn or f(x) = 1 * x-n.
Applying the quotient rule, which states that the derivative of a quotient is given by (v*u' - u*v')/v2, where u = 1 and v = xn. Therefore:
- u' (derivative of 1) = 0
- v' (derivative of xn) = n * xn-1
- Inserting these into the quotient rule formula, we get: (xn * 0 - 1 * n * xn-1) / x2n
- Simplifying, we have: -n * xn-1 / x2n
- This further simplifies to: -n * x-n-1, confirming the power rule for negative exponents.
The above demonstrates that using the quotient rule, we can prove the validity of the power rule for negative whole number exponents, where a function with a negative exponent when differentiated yields the exponent multiplied by the function with the exponent decreased by one.