Derivative of x^(1/x)

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

The key concepts are mentioned below:

Power Function: x^(1/x) can be rewritten as e^(ln(x)/x).

Chain Rule: Rule for differentiating functions like e^(ln(x)/x).

Logarithmic Differentiation: Useful for differentiating functions like x^(1/x).

The derivative of x^(1/x) can be denoted as d/dx (x^(1/x)) or (x^(1/x))'.

The formula we use to differentiate x^(1/x) involves logarithmic differentiation: d/dx (x^(1/x)) = x^(1/x) * [(1 - ln(x))x²]

This formula applies to all x > 0.

We can derive the derivative of x^(1/x) using proofs. To show this, we will use logarithmic differentiation along with the rules of differentiation.

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

• By Logarithmic Differentiation
   
• Using Chain Rule
   
• Using Quotient Rule

We will now demonstrate that the differentiation of x^(1/x) results in x^(1/x) * [(1 - ln(x))x²] using the above-mentioned methods:

By Logarithmic Differentiation

The derivative of x^(1/x) can be proved using logarithmic differentiation, which simplifies the differentiation process for functions raised to variable exponents. To find the derivative of x^(1/x), consider y = x^(1/x). Taking the natural logarithm on both sides, ln(y) = (1/x) * ln(x) Differentiating both sides with respect to x, 1/y * dy/dx = (d/dx (ln(x))x) = (1/x²) - (ln(x)x²) dy/dx = y * [(1 - ln(x))x²] Since y = x^(1/x), substitute back to get the derivative: dy/dx = x^(1/x) * [(1 - ln(x))x²]

Using Chain Rule

To prove the differentiation of x^(1/x) using the chain rule, We use the transformation: x^(1/x) = e^(ln(x)/x) Consider u(x) = ln(x)x, then y = e^u By chain rule: dy/dx = e^u * du/dx du/dx = (1/x²) - (ln(x)/x²) Thus, dy/dx = e^(ln(x)/x) * [(1 - ln(x))/x²] Since e^(ln(x)/x) = x^(1/x), dy/dx = x^(1/x) * [(1 - ln(x))/x²]

Using Quotient Rule

We can rewrite the function as e^(ln(x)/x) for differentiation. Let u = ln(x) and v = x Therefore, y = u/v Using the quotient rule: d/dx (u/v) = (v * du/dx - u * dv/dx)/v² du/dx = 1/x and dv/dx = 1 Applying to the quotient rule: (1 * 1/x - ln(x) * 1)/x² = (1 - ln(x))/x² Thus, the derivative is: dy/dx = x^(1/x) * [(1 - ln(x))/x²]

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^(1/x).

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^(1/x), 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 function x^(1/x) is not defined, and thus its derivative is also undefined.

For x = 1, the derivative of x^(1/x) simplifies to 1, as x^(1/x) = 1 and the derivative of a constant is 0.

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

• Logarithmic Differentiation: A method used to differentiate functions with variable exponents by taking the natural log of both sides.

• Chain Rule: A rule for differentiating compositions of functions.

• Power Function: A function where the variable is raised to a power, such as x^(1/x).

• Quotient Rule: A rule for differentiating the ratio of two functions.