Derivative of 1 over x
2026-02-28 13:49 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-1-over-x

We can derive the derivative of 1/x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:

  1. By First Principle
  2. Using Power Rule
  3. Using Quotient Rule

We will now demonstrate that the differentiation of 1/x results in -1/x² using the above-mentioned methods:

By First Principle

The derivative of 1/x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 1/x using the first principle, we will consider f(x) = 1/x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = 1/x, we write f(x + h) = 1/(x + h).

Substituting these into equation (1),

f'(x) = limₕ→₀ [1/(x + h) - 1/x] / h = limₕ→₀ [(x - (x + h)) / (x(x + h))] / h = limₕ→₀ [-h / (x² + xh)] / h = limₕ→₀ -1 / (x² + xh) = -1/x²

Hence, proved.

Using Power Rule

To prove the differentiation of 1/x using the power rule, We rewrite 1/x as x⁻¹.

By the power rule: d/dx [xⁿ] = nxⁿ⁻¹ Let n = -1, d/dx (x⁻¹) = -1 * x⁻² = -1/x²

Using Quotient Rule

We will now prove the derivative of 1/x using the quotient rule. The step-by-step process is demonstrated below:

Here, 1/x can be viewed as (1)/(x)

Using the quotient rule formula:

d/dx [u/v] = (v * u' - u * v') / v²

Let u = 1 (constant), v = x u' = 0, v' = 1

d/dx (1/x) = (x * 0 - 1 * 1) / x² = -1/x²

Thus, d/dx (1/x) = -1/x².