What is the derivative of x to the power x? This question may seem straightforward, but it actually leads to a fascinating and complex mathematical journey. The function f(x) = x^x is known as the exponential function with base e, and its derivative is a topic of great interest in calculus and mathematical analysis.
The function f(x) = x^x can be rewritten as e^(xln(x)), where ln(x) is the natural logarithm of x. This transformation is useful because it allows us to apply the chain rule and the product rule to find the derivative. Let’s explore this process step by step.
First, we need to find the derivative of the function g(x) = xln(x). To do this, we can use the product rule, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. In this case, we have:
g'(x) = (x)’ln(x) + x(ln(x))’
g'(x) = 1ln(x) + x(1/x)
g'(x) = ln(x) + 1
Now that we have the derivative of g(x), we can find the derivative of f(x) = e^(xln(x)) using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, we have:
f'(x) = (e^(xln(x)))’ (xln(x))’
f'(x) = e^(xln(x)) (ln(x) + 1)
Since e^(xln(x)) = x^x, we can rewrite the derivative as:
f'(x) = x^x (ln(x) + 1)
This is the derivative of the function f(x) = x^x. It is a combination of the function itself and its natural logarithm, which makes it a non-elementary function. This means that it cannot be expressed in terms of a finite combination of elementary functions, such as polynomials, exponentials, logarithms, and trigonometric functions.
The derivative of x to the power x is an intriguing example of how calculus can lead to complex and non-trivial results. It highlights the beauty and depth of mathematical analysis and demonstrates the power of mathematical tools like the chain rule and the product rule. By exploring this derivative, we gain a deeper understanding of the behavior of exponential functions and the intricacies of calculus.
