Derivative of sqrt(3x)
2026-02-28 06:18 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-sqrt(3x)

We can derive the derivative of sqrt(3x) using proofs. To show this, we will use the rules of differentiation and chain rule.

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

  1. By First Principle
  2. Using Chain Rule

We will now demonstrate that the differentiation of sqrt(3x) results in 3/(2sqrt(3x)) using the above-mentioned methods:

By First Principle

The derivative of sqrt(3x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of sqrt(3x) using the first principle, we will consider f(x) = sqrt(3x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = sqrt(3x), we write f(x + h) = sqrt(3(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [sqrt(3(x + h)) - sqrt(3x)] / h

Multiply and divide by the conjugate: = limₕ→₀ [(sqrt(3(x + h)) - sqrt(3x)) * (sqrt(3(x + h)) + sqrt(3x))] / [h * (sqrt(3(x + h)) + sqrt(3x))] = limₕ→₀ [3(x + h) - 3x] / [h * (sqrt(3(x + h)) + sqrt(3x))] = limₕ→₀ [3h] / [h * (sqrt(3(x + h)) + sqrt(3x))]

Cancel h: = limₕ→₀ 3 / [sqrt(3(x + h)) + sqrt(3x)] = 3 / [2sqrt(3x)]

Hence, proved.

Using Chain Rule

To prove the differentiation of sqrt(3x) using the chain rule, We use the formula: Sqrt(3x) = (3x)^(1/2) Let u = 3x

Then, sqrt(3x) = u^(1/2)

By the chain rule: d/dx [u^(n)] = n * u^(n-1) * (du/dx)… (1)

Let’s substitute u = 3x, n = 1/2 into equation (1), d/dx (sqrt(3x)) = (1/2) * (3x)^(-1/2) * d/dx (3x) = (1/2) * (3x)^(-1/2) * 3 = 3/(2sqrt(3x))

Thus: d/dx (sqrt(3x)) = 3/(2sqrt(3x)).