Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,
d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹
where m is a positive integer.

Guida verificata passo dopo passo

Start by expressing the function x^(-m) as a fraction: x^(-m) = 1/x^m.

Apply the Quotient Rule for derivatives, which states that if you have a function in the form of u/v, the derivative is (v * du/dx - u * dv/dx) / v^2.

Set u = 1 and v = x^m. Then, compute the derivatives: du/dx = 0 and dv/dx = m * x^(m-1).

Substitute these into the Quotient Rule formula: (x^m * 0 - 1 * m * x^(m-1)) / (x^m)^2.

Simplify the expression: -m * x^(m-1) / x^(2m) = -m * x^(-m-1), which proves the Power Rule for negative integers.

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