Derivative of Root 2x

To find the derivative of √(2x), we represent it as d/dx (√(2x)) or (√(2x))'. The derivative is 1/√(2x), indicating that the function is differentiable within its domain.

The key concepts involved include: -

Square Root Function: √(2x). 

Chain Rule: Used for differentiating composite functions like √(2x). 

Derivative of x^n: The general rule for differentiating functions with exponents.

The derivative of √(2x) can be denoted as d/dx (√(2x)) or (√(2x))'. The formula we use to differentiate √(2x) is: d/dx (√(2x)) = 1/√(2x) The formula is applicable for all x > 0, where the square root function is defined.

We can derive the derivative of √(2x) using proofs. To show this, we utilize the rules of differentiation. Here are some methods used to prove this:

By First Principle

The derivative of √(2x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

Consider f(x) = √(2x). Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given f(x) = √(2x), then f(x + h) = √(2(x + h)).

Substituting these into the equation: f'(x) = limₕ→₀ [√(2(x + h)) - √(2x)] / h Using the identity a - b = (a^2 - b^2)/(a + b)

for simplification: = limₕ→₀ (2(x + h) - 2x) / [h(√(2(x + h)) + √(2x))] = limₕ→₀ [2h] / [h(√(2(x + h)) + √(2x))] = limₕ→₀ 2 / (√(2(x + h)) + √(2x))

As h approaches 0, we get: f'(x) = 2 / (2√(2x)) = 1/√(2x)

Hence, proved.

Using Chain Rule

To prove the differentiation of √(2x) using the chain rule,

let: y = √(2x) = (2x)^(1/2)

By the chain rule: d/dx (2x)^(1/2) = (1/2)(2x)^(-1/2) * (d/dx (2x)) = (1/2)(2x)^(-1/2) * 2 = 1/√(2x)

When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a bit complex.

To understand them better, consider a scenario where the speed changes (first derivative), and the rate at which the speed changes (second derivative) also varies. Higher-order derivatives make it easier to understand functions like √(2x).

For the first derivative of a function, we write f′(x), indicating how the function changes or its slope at a specific point. The second derivative is derived from the first derivative, denoted as f′′(x). Similarly, the third derivative, f′′′(x), results from the second derivative, and this pattern continues.

For the nth Derivative of √(2x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change.

When x is 0, the derivative is undefined because √(2x) has an undefined point at x = 0. For any positive x, the derivative of √(2x) = 1/√(2x), which varies with different values of x.

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

• Square Root Function: A function that involves the square root of a variable, such as √(2x).

• Chain Rule: A rule for differentiating composite functions.

• Quotient Rule: A rule used to differentiate ratios of two functions.

• Undefined Point: A value of x for which a function does not produce a valid result.