Derivative of Root X

Square root differentiation is the process of finding the derivative of square root functions. The derivative of √x is represented by d/dx(√x) or d/dx(√x^½). It involves the process of finding the change of the square root function in respect to the change of the fundamental variable. For √x, the derivative is 1/2√x which represents how the curve of the square root function changes as the value of x changes.

To find the derivative of root x the formula is given below:
    
        d/dx (√x) = 1/2√x.

The derivative of root x can be proved using various approaches like using the definition of a derivative (first principle rule), using the chain rule, and using the product rule. Let us now see how the various rules are used in detail:

The step-by-step way to find the derivative of root x using the first principle rule is given below:

 

Step 1: Recall the First Principle Formula:

The derivative of a function f(x) using the first principle rule is given by
            
 f’(x) = h→0 f(x + h) - f(x)/h

 Here, h is the small change in x.

Step 2: Substitute f(x) = √x into the formula:

For f(x) = √x substitute this into the formula:

f’(x) = h→0 √x + h  -  x/h

Step 3: Simplify the Numerator:

The numerator √x + h  - √x contains square roots, so multiply and divide by the    conjugate to eliminate the square roots:

√x + h  -  √x/h x √x + h  +  √x/√x + h  +  √x = (x + h) - x/h(√x + h + √x
        
 Simplify the numerator: 
        
 (x + h) - x = h.

Step 4: Simplify the Equation:

The fraction now becomes:        
            
 f’(x) = h→0 h/h√x + h + √x

Since h is not equal to zero, cancel h from numerator and denominator
        
 f’(x) = h→0 1x + h + x

Step 5: Take the Limit as h → 0:

As h → 0, √x + h → √x. Substitute this into the equation:

f’(x) = 1/√x + √x  = 1/2√x

Final Answer: 

The derivative of the root of x using the first principle is:
        
 f’(x) = 1/2√x

To find the Formula of derivative of the root of x using the chain rule, we follow the steps given below:

Step 1: Rewrite x  in Exponent Form:
The root of x can be written as:
        
            x = x1/2.

Step 2: Recognize the Chain Rule:
The chain rule states: 

            ddx[f(g(x))] = f’(g(x)) x g’(x).

    Here, we apply this rule to functions that can be expressed in a composite form

    Step 3: Apply the chain rule to x1/2
In the case of x = x1/2, there is no inner function to simplify further, since g(x) = x is the simplest form. Hence, we directly apply the power rule, which is used in special cases of chain rule.

Step 4: Differentiate using the power rule:
The power rule states: 

            ddx[xn] = n x xn-1

Substitute n = ½ into the power rule:

ddx[x1/2] = ½ x x1/2 - 1 

    
    Step 5: Simplify the Exponent:
            
                ½ x x1/2 - 1 = ½ x x-1/2

Step 6: Rewrite in the terms of root:
Using the property x-½     = 1x, we can write it as:
        
            ½ x x-½  = 12x

    Final Answer:
    The derivative of x is:
    
                ddx(x) = 12x

The product rule is used to differentiate a function that is a product of two different functions. The product rule states that:

            ddx[u(x) x v(x)] = u’(x)v(x) + u(x)v’(x).

To find the derivative of the root of x using the product rule, the step-by-step procedure is given below:

Step 1: Rewrite x as a product:
Recall that x = x1/2 we can express this as a product:
        
            x = x1/2  = x1/2 x 1

Here, we consider u(x) = x1/2 and v(x) = 1. This will allow us to apply the product rule.

Step 2: Identify u(x) and v(x):
    Let us take:
        
                u(x) = x1/2 and v(x) = 1

Step 3: Differentiate u(x) and v(x):
The derivative of u(x) = x1/2 is:
        
        u’(x) = 12x-1/2 = 12x

The derivative of v(x) = 1 is:
        
        v’(x) = 0

    Step 4: Apply the product rule:
    Substitute the values to the given formula
        
                ddx[u(x) x v(x)] = u’(x)v(x) + u(x)v’(x).
    
    This becomes:
        
                ddx[x] = (12x)(1) + (x1/2)(0)

    Step 5: Simplify the Equation:
    The second term becomes 0 as v’(x) = 0. Hence;
                ddx[x] = 12x

    Final Answer:
    The derivative of the root of x using the product rule is:

                ddx[x] = 12x

The higher-order derivatives of the root of x involves finding the derivative of its first derivative and contributing to differentiate successively to obtain a more complex equation. These higher derivatives provide insights into the behavior of the root function and are useful in various applications like mathematical modeling and series expansions.

Nth Derivative of the Root of X

To find the Nth derivative of the root of x, we express it by finding a more convenient form of differentiation. The root or x can be written as x1/2. The general formula used to find the Nth derivative of the root of x is given below

        f(N)(x) = (-1)N-1  * (2N-3)!!2N x x(1/2) - N
Where (2N - 3)!! Is double factorial.

Special Cases:

For N = 0:
The 0th derivative is just the original function of x1/2.
For N = 1:
The first derivative is 12x-1/2
For higher derivatives:
Each differentiation reduces the exponent by 1. As N increases, the coefficients become more complex.
When x = 0:
The Nth derivative for x1/2 at x = 0 is undefined.

• Derivatives: It is a measure of a function changes as its input changes. In geometry, it is the slope of a tangent line to the functions graph.

• Slope: A slope is the measure of the steepness of a line. It shows the change in y divided by the change in x.

• Tangent Line: A tangent line is a line that touches a curve at a single point and has the same slope as the curve at that point.

• Domain: A domain is the set of all possible input values (x) for with a function is defined.