Rivalry2

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Original 2026-01-01

Modified 2026-02-28

1 We can derive the derivative of x²/2 using proofs.

2 To show this, we will use the rules of differentiation.

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

4 By First Principle

5 Using Power Rule

6 Using Constant Multiple Rule

7 We will now demonstrate that the differentiation of x²/2 results in x using the above-mentioned methods:

8 By First Principle

9 The derivative of x²/2 can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.

10 To find the derivative of x²/2 using the first principle, we will consider f(x) = x²/2.

11 Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

12 Given that f(x) = x²/2, we write f(x + h) = (x + h)²/2.

13 Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [(x + h)²/2 - x²/2] / h = limₕ→₀ [(x² + 2xh + h²)/2 - x²/2] / h = limₕ→₀ [2xh + h²]/2h = limₕ→₀ [x + h/2] As h approaches 0, f'(x) = x.

14 Hence, proved.

15 Using Power Rule

16 To prove the differentiation of x²/2 using the<a>power</a>rule, We use the formula: d/dx (xⁿ) = n*xⁿ⁻¹

17 For x², n = 2.

18 So, d/dx (x²) = 2*x¹ = 2x. Since we have x²/2, d/dx (x²/2) = (1/2)*d/dx (x²) = (1/2)*2x = x.

19 Using Constant Multiple Rule

20 We will now prove the derivative of x²/2 using the constant<a>multiple</a>rule.

21 The formula we use is: d/dx (c*f(x)) = c*d/dx (f(x))

22 Let c = 1/2 and f(x) = x².

23 So, d/dx (x²/2) = (1/2)*d/dx (x²) = (1/2)*2x = x.