Derivative of 1/(x+2)
2026-02-28 06:07 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-1/(x+2)

We can derive the derivative of 1/(x+2) using proofs. To show this, we will use algebraic manipulation along with the rules of differentiation.

There are several methods we use to prove this, such as:

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

We will now demonstrate that the differentiation of 1/(x+2) results in -1/(x+2)² using the above-mentioned methods:

By First Principle

The derivative of 1/(x+2) 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+2) using the first principle, we will consider f(x) = 1/(x+2). 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+2), we write f(x + h) = 1/(x+h+2).

Substituting these into equation (1), f'(x) = limₕ→₀ [1/(x+h+2) - 1/(x+2)] / h = limₕ→₀ [(x+2) - (x+h+2)] / [(x+2)(x+h+2)h] = limₕ→₀ [-h] / [(x+2)(x+h+2)h] = limₕ→₀ -1 / [(x+2)(x+h+2)]

As h approaches 0, we simplify the expression: f'(x) = -1 / (x+2)²Hence, proved.

Using Chain Rule

To prove the differentiation of 1/(x+2) using the chain rule, We use the formula: 1/(x+2) = (x+2)⁻¹

Let u = x+2

Thus, y = u⁻¹

Using the chain rule formula: d/dx [uⁿ] = nuⁿ⁻¹ du/dx dy/dx = -1 * (x+2)⁻² * d/dx (x+2) dy/dx = -1/(x+2)² Hence, proved.

Using Power Rule

We will now prove the derivative of 1/(x+2) using the power rule. The step-by-step process is demonstrated below:

Rewrite the function: y = (x+2)⁻¹

Using the power rule to differentiate: d/dx (y) = -1 * (x+2)⁻² * d/dx (x+2) d/dx (y) = -1/(x+2)²

Thus, the derivative of 1/(x+2) is -1/(x+2)².