Derivative of x^3/2

We now understand the derivative of x^(3/2). It is commonly represented as d/dx (x^(3/2)) or (x^(3/2))', and its value is (3/2)x^(1/2). The function x^(3/2) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Power Function: (x^(3/2)).

Power Rule: Rule for differentiating functions of the form xⁿ.

Square Root Function: Results in the derivative having a square root form.

The derivative of x^(3/2) can be denoted as d/dx (x^(3/2)) or (x^(3/2))'.

The formula we use to differentiate x^(3/2) is: d/dx (x^(3/2)) = (3/2)x^(1/2) (or) (x^(3/2))' = (3/2)x^(1/2)

The formula applies to all x where x > 0.

We can derive the derivative of x^(3/2) using proofs. To show this, we will use the power rule along with basic differentiation rules.

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

• By First Principle
   
• Using Power Rule
   
• Using Chain Rule

We will now demonstrate that the differentiation of x^(3/2) results in (3/2)x^(1/2) using the above-mentioned methods:

By First Principle

The derivative of x^(3/2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x^(3/2) using the first principle, we will consider f(x) = x^(3/2). 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^(3/2), we write f(x + h) = (x + h)^(3/2). Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)^(3/2) - x^(3/2)] / h = limₕ→₀ [(x^(3/2) + (3/2)x^(1/2)h + ...) - x^(3/2)] / h = limₕ→₀ [(3/2)x^(1/2)h + ...] / h Simplifying the terms: f'(x) = limₕ→₀ (3/2)x^(1/2) + ... As h approaches 0, the higher-order terms vanish, leaving: f'(x) = (3/2)x^(1/2) Hence, proved.

Using Power Rule

To prove the differentiation of x^(3/2) using the power rule, We use the formula: d/dx (x^n) = nx^(n-1) For x^(3/2), let n = 3/2. So, d/dx (x^(3/2)) = (3/2)x^{(3/2 - 1)} = (3/2)x^(1/2) Thus, using the power rule, we get the derivative as (3/2)x^(1/2).

Using Chain Rule

We will now prove the derivative of x^(3/2) using the chain rule. The step-by-step process is demonstrated below: Here, we use the expression: x^(3/2) = (x¹)^(3/2) Let u = x, then u^(3/2) = x^(3/2) The derivative using the chain rule: d/dx (u^(3/2)) = (3/2)u^(1/2) * du/dx Since u = x, du/dx = 1 Therefore, d/dx (x^(3/2)) = (3/2)x^(1/2) Thus, the derivative is (3/2)x^(1/2).

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^(3/2).

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^(3/2), we generally use f n(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 is not defined because x^(3/2) is not differentiable at x = 0.

For positive values of x, the derivative of x^(3/2) is positive, indicating an increasing function.

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

• Power Rule: A fundamental rule in calculus used to differentiate functions of the form x^n.

• Chain Rule: A method in calculus for finding the derivative of the composition of two or more functions.

• Product Rule: A rule used to find the derivative of the product of two functions.

• Square Root Function: A function that involves the square root of a variable, often appearing in derivatives of fractional powers. ```