Derivative of 1/t
2026-02-28 06:03 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-1/t

We can derive the derivative of 1/t 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/t results in -1/t² using the above-mentioned methods:

By First Principle

The derivative of 1/t can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 1/t using the first principle, we will consider f(t) = 1/t. Its derivative can be expressed as the following limit.

f'(t) = limₕ→₀ [f(t + h) - f(t)] / h … (1)

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

Substituting these into equation (1), f'(t) = limₕ→₀ [1/(t + h) - 1/t] / h = limₕ→₀ [(t - (t + h)) / (t(t + h))] / h = limₕ→₀ [-h / (t² + th)] / h = limₕ→₀ [-1 / (t² + th)]

As h approaches 0, we get: f'(t) = -1/t². Hence, proved.

Using Power Rule

To prove the differentiation of 1/t using the power rule, We rewrite 1/t as t⁻¹. d/dt (t⁻¹) = -1 * t⁻² Since t⁻² = 1/t², d/dt (1/t) = -1/t².

Using Quotient Rule We will now prove the derivative of 1/t

using the quotient rule.

The step-by-step process is demonstrated below: Here, we use the formula, 1/t = 1 * (t)⁻¹ Given that, u = 1 and v = t Using the quotient rule formula: d/dt [u/v] = (v * u' - u * v') / v² u' = d/dt (1) = 0 (substitute u = 1) v' = d/dt (t) = 1 (substitute v = t)

Again, use the quotient rule formula: d/dt (1/t) = (t * 0 - 1 * 1) / t² = -1/t²