Derivative of x²/2

We now understand the derivative of x²/2.

It is commonly represented as d/dx (x²/2) or (x²/2)', and its value is x.

The function x²/2 has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Power Function: (x²/2 is a polynomial function).

Power Rule: Rule for differentiating x²/2.

Constant Multiple Rule: The derivative of a constant times a function.

The derivative of x²/2 can be denoted as d/dx (x²/2) or (x²/2)'. The formula we use to differentiate x²/2 is: d/dx (x²/2) = x (x²/2)' = x The formula applies to all x.

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.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.

To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes.

Higher-order derivatives make it easier to understand functions like x²/2.

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.

The second derivative is derived from the first derivative, which is denoted using f′′(x).

Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues.

For the nth Derivative of x²/2, we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change (continuing for higher-order derivatives).

When x is 0, the derivative of x²/2 = 0, which means there is no change at this point. When x is negative, the derivative of x²/2 is negative, indicating a downward slope.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Power Rule: A differentiation rule used to find the derivative of a power function.

• Constant Multiple Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

• First Derivative: The initial result of differentiating a function, indicating the rate of change of the function.

• Quotient Rule: A rule used to differentiate functions that are divided by each other.