Derivative of x²/2
2026-02-28 23:51 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-x%5E2/2

We can derive the derivative of x²/2 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 Constant Multiple Rule

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

By First Principle

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

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

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

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

Substituting these into equation (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.

Hence, proved.

Using Power Rule

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

For x², n = 2.

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.

Using Constant Multiple Rule

We will now prove the derivative of x²/2 using the constant multiple rule.

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

Let c = 1/2 and f(x) = x².

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