Derivative of 1/u
2026-02-28 06:08 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-1/u

We can derive the derivative of 1/u using proofs. To show this, we will use algebraic identities along with 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/u results in -1/u² using the above-mentioned methods:

By First Principle

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

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

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

Substituting these into equation (1), f'(u) = limₕ→₀ [1/(u + h) - 1/u] / h = limₕ→₀ [u - (u + h)] / [h(u + h)u] = limₕ→₀ [-h] / [h(u + h)u] = limₕ→₀ -1 / [(u + h)u]

As h approaches 0, f'(u) = -1/u²

Hence, proved.

Using Power Rule

To prove the differentiation of 1/u using the power rule, We rewrite the function: 1/u = u⁻¹

Using the power rule: d/du (uⁿ) = n·uⁿ⁻¹ d/du (u⁻¹) = -1·u⁻² Thus, d/du (1/u) = -1/u²

Using Quotient Rule

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

Here, we use the identity, 1/u = 1 / u

Given that, f(u) = 1 and g(u) = u

Using the quotient rule formula: d/du [f(u) / g(u)] = [f '(u) g(u) - f(u) g'(u)] / [g(u)]²… (1)

Let’s substitute f(u) = 1 and g(u) = u in equation (1), d/du (1/u) = [0·u - 1·1] / u² = -1/u²

Thus, d/du (1/u) = -1/u².