Question

Use the reciprocal rule to verify that the power rule is valid for negative integers; that is, d/dx[x⁻ⁿ] = -nx⁻ⁿ-1 for all positive integers n.

a) True
b) False

Answer 1

The reciprocal rule can be used to verify the power rule for negative integers.

Step-by-step explanation:

The reciprocal rule states that the derivative of x⁻ⁿ is equal to -n*x⁻ⁿ⁻¹. We can use this rule to verify that the power rule is valid for negative integers. Let's differentiate the function f(x) = x⁻ⁿ using the reciprocal rule:

d/dx[x⁻ⁿ] = d/dx[(1/xⁿ)] = -1/xⁿ⁺¹ = -x⁻ⁿ⁻¹

Therefore, we have verified that the power rule holds true for negative integers.