Derivative of 5/x
2026-02-28 06:19 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-5/x

We can derive the derivative of 5/x 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:

By First Principle

Using Power Rule

Using Quotient Rule

We will now demonstrate that the differentiation of 5/x results in -5/x² using the above-mentioned methods:

By First Principle

The derivative of 5/x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 5/x using the first principle, we will consider f(x) = 5/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) = 5/x, we write f(x + h) = 5/(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [5/(x + h) - 5/x] / h = limₕ→₀ [5x - 5(x + h)] / [h(x)(x + h)] = limₕ→₀ [-5h] / [h(x)(x + h)] = limₕ→₀ [-5] / [x(x + h)] = -5/x²

Hence, proved.

Using Power Rule

To prove the differentiation of 5/x using the power rule, We rewrite the function as: 5/x = 5x-1.

Using the power rule, if y = xn, then dy/dx = nx(n-1).

So we get, d/dx (5x-1) = 5(-1)x(-1-1) = -5x-2.

Therefore, d/dx (5/x) = -5/x².

Using Quotient Rule

We will now prove the derivative of 5/x using the quotient rule.

The step-by-step process is demonstrated below:

Here, we use the formula, 5/x = 5 * (1/x).

Given that u = 5 and v = x⁻¹,

Using the quotient rule formula: d/dx [u/v] = [u'v - uv']/v². u' = 0 (since u = 5 is a constant) v' = d/dx (x⁻¹) = -x⁻² (substitute v = x⁻¹).

Using the quotient rule, d/dx (5/x) = [0 * x⁻¹ - 5 * (-x⁻²)] / (x⁻¹)² = (5x⁻²)/x⁻² = -5/x².

Hence, d/dx (5/x) = -5/x².