Derivative of 3/x
2026-02-28 06:16 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-3/x

We can derive the derivative of 3/x using proofs. To show this, we will use the rules of differentiation.

There are several methods we use to prove this, such as: By First Principle Using Power Rule Using Product Rule

We will now demonstrate that the differentiation of 3/x results in -3/x² using the above-mentioned methods: By First Principle The derivative of 3/x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 3/x using the first principle, we will consider f(x) = 3/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) = 3/x, we write f(x + h) = 3/(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [3/(x + h) - 3/x] / h = limₕ→₀ [3x - 3(x + h)] / [x(x + h)h] = limₕ→₀ [-3h] / [x(x + h)h] = limₕ→₀ -3 / [x(x + h)] = -3 / x² Hence, the derivative of 3/x is -3/x².

Using Power Rule To prove the differentiation of 3/x using the power rule, We express 3/x as 3x⁻¹.

Using the power rule, d/dx [x^n] = n*x^(n-1), d/dx [3x⁻¹] = -1*3x^(-1-1) = -3x⁻² = -3/x²

Thus, the derivative of 3/x is -3/x².

Using Product Rule We will now prove the derivative of 3/x using the product rule.

The step-by-step process is demonstrated below: Here, we use the formula, 3/x = 3 * x⁻¹ Let u = 3 and v = x⁻¹

Using the product rule formula: d/dx [u*v] = u'v + uv' u' = d/dx (3) = 0 v' = d/dx (x⁻¹) = -1*x⁻² d/dx (3/x) = 0 * x⁻¹ + 3 * (-1)x⁻² = -3x⁻² = -3/x²

Therefore, the derivative of 3/x is -3/x².