Derivative of (x-2)^2

We now understand the derivative of (x-2)². It is commonly represented as d/dx ((x-2)²) or ((x-2)²)', and its value is 2(x-2). The function (x-2)² has a clearly defined derivative, indicating it is differentiable across its domain.

The key concepts are mentioned below:

Polynomial Function: A standard form of algebraic expression, such as (x-2)².

Power Rule: A basic rule for differentiating polynomial functions.

Derivative: Represents the rate of change of a function.

The derivative of (x-2)² can be denoted as d/dx ((x-2)²) or ((x-2)²)'.

The formula we use to differentiate (x-2)² is: d/dx ((x-2)²) = 2(x-2)

The formula applies to all x.

We can derive the derivative of (x-2)² using proofs. To show this, we will use basic differentiation rules.

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

• By First Principle
   
• Using Power Rule
   
• Using Expansion

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

By First Principle

The derivative of (x-2)² can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of (x-2)² using the first principle, we will consider f(x) = (x-2)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) = (x-2)², we write f(x + h) = (x + h - 2)². Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h - 2)² - (x-2)²] / h = limₕ→₀ [(x + h - 2)(x + h - 2) - (x-2)(x-2)] / h = limₕ→₀ [x² + 2xh + h² - 4x - 4h + 4 - (x² - 4x + 4)] / h = limₕ→₀ [2xh + h² - 4h] / h = limₕ→₀ [h(2x + h - 4)] / h Cancel h and take the limit as h approaches 0, f'(x) = 2x - 4 Thus, f'(x) = 2(x-2). Hence, proved.

Using Power Rule

To prove the differentiation of (x-2)² using the power rule, We use the formula: d/dx [uⁿ] = n * u^(n-1) * du/dx Consider u = (x-2) and n = 2 Then, d/dx ((x-2)²) = 2 * (x-2)^(2-1) * d/dx(x-2) = 2(x-2) * 1 = 2(x-2)

Using Expansion

We will now prove the derivative of (x-2)² using expansion. The step-by-step process is demonstrated below: Expand (x-2)² to get x² - 4x + 4. Differentiate each term separately: d/dx(x²) = 2x d/dx(-4x) = -4 d/dx(4) = 0 Combine the derivatives: f'(x) = 2x - 4 Thus: f'(x) = 2(x-2).

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 (x-2)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 (x-2)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).

For any constant shift in x, the derivative of (x-2)² is simply recalculated using the same formula, resulting in 2(x-2).

At x = 2, the derivative of (x-2)² = 2(2-2) = 0, indicating a horizontal tangent at this point.

• Derivative: A measure of how a function changes as its input changes.

• Polynomial Function: An algebraic expression consisting of variables and coefficients.

• Power Rule: A rule in calculus for differentiating expressions of the form xⁿ.

• First Principle: A method of finding the derivative based on limits.

• Chain Rule: A formula for computing the derivative of the composition of two or more functions.