Derivative of x/3
2026-02-28 06:14 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-x/3

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

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

We will now demonstrate that the differentiation of x/3 results in 1/3 using the above-mentioned methods: By First Principle The derivative of x/3 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

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

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

Using Constant Rule To prove the differentiation of x/3 using the constant rule, We use the formula: d/dx (c·x) = c Here, c = 1/3 Therefore, d/dx (x/3) = 1/3.

Using Power Rule We will now prove the derivative of x/3 using the power rule.

The step-by-step process is demonstrated below: Consider f(x) = x/3 = (1/3)x.

Using the power rule formula: d/dx [x^n] = n·x^(n-1) Here, n = 1. d/dx (x/3) = d/dx [(1/3)x] = (1/3)·d/dx [x^1] = (1/3)·1·x^(1-1) = 1/3

Thus, the derivative of x/3 is 1/3.