Question

Evaluate the following derivatives. $$\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right)$$

Short answer

Question: Find the derivative of the function $$f(x) = x^{\left(x^{10}\right)}$$ with respect to x. Answer: The derivative of the function $$f(x) = x^{\left(x^{10}\right)}$$ with respect to x is $$\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right) = x^{10}x^{\left(x^{10}-1\right)} + 10x^{9}x^{x^{10}}$$.

Step-by-step solution

Identify the outer and inner functions

The given function is of the form $$u^v$$, where $$u = x$$ and $$v = x^{10}$$. We will need to find the derivatives of both these functions and then apply the chain rule to find the overall derivative.

Differentiate the outer function

The outer function is $$u^v$$. To differentiate it with respect to x, we will use the chain rule. The chain rule states that if u and v are functions of x, then: $$\frac{d}{d x}(u^v) = vu^{(v-1)}\frac{d u}{d x}$$ Differentiating $$u = x$$ with respect to x, we get: $$\frac{d u}{d x} = 1$$

Differentiate the inner function

The inner function is $$v = x^{10}$$. To differentiate it with respect to x, we will use the power rule which states that if $$v(x) = x^n$$, then: $$\frac{d v}{d x} = nx^{(n-1)}$$ Differentiating $$v = x^{10}$$ with respect to x, we get: $$\frac{d v}{d x} = 10x^{9}$$

Apply the chain rule

Using the chain rule, we have that, $$\frac{d}{d x}(u^v) = vu^{(v-1)}\frac{d u}{d x} + u^v\frac{d v}{d x}$$ Now, substitute the expressions for u, v and their derivatives from previous steps into the chain rule formula: $$\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right) = x^{10}x^{\left(x^{10}-1\right)}(1) + x^{x^{10}}(10x^{9})$$

Simplify the expression

Combine the terms and simplify the expression for the derivative: $$\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right) = x^{10}x^{\left(x^{10}-1\right)} + 10x^{9}x^{x^{10}}$$ This is the derivative of the given function.

Key concepts

Derivative of Exponential Functions

Understanding the derivative of exponential functions is essential when dealing with complex mathematical expressions. Typically, an exponential function is defined as any function in which a variable appears in the exponent. For example, the function given by \(a^x\), where \(a\) is a constant and \(x\) is the exponent.
For the general exponential function \(a^x\), the derivative with respect to \(x\) is \(a^x\ln(a)\), assuming \(a\) is a positive constant. But what happens when the exponent itself is a function of \(x\), like \(x^{f(x)}\)? In this case, we must use logarithmic differentiation, which involves taking the natural logarithm of both sides of the equation to expose the exponent, and then differentiating implicitly. The procedure simplifies the expression so the power rule can be applied and is a key technique in handling more complex exponential derivatives, especially when both the base and the exponent are functions of \(x\).

Power Rule

The power rule is a basic differentiation technique often used as the first step when finding derivatives. It states that if you have a monomial \(x^n\), where \(n\) is a real number, the derivative of this function with respect to \(x\) is \(nx^{n-1}\).
This rule simplifies the process of differentiation because it eliminates the need for the limit definition of the derivative. The rule is applicable to both integer and fractional exponents, making it a versatile tool in a mathematician's toolkit. When the exponent is a function of \(x\), as in the exercise above, the power rule provides the basis for the subsequent application of the chain rule.

Applying Chain Rule in Calculus

The chain rule is a fundamental differentiation technique in calculus used when dealing with composite functions, where one function is nested inside another. When you have a function \(f(g(x))\), the chain rule tells us how to find the derivative by multiplying the derivative of the outer function \(f\) with respect to the inner function \(g\), by the derivative of the inner function \(g\) with respect to \(x\).
In mathematical terms, if \(y=f(u)\) and \(u=g(x)\), then:\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\].
The chain rule is invaluable when functions are not merely stacked together but are also intertwined, as seen in the original exercise where \(x\) is raised to the power of \(x^{10}\). We differentiate the outer function while treating the inner function as a constant, then multiply the result by the derivative of the inner function. It allows us to elegantly handle the complexities of such derivatives without expanding the function into a more cumbersome form.

Differentiation Techniques

There are several differentiation techniques that can be used when the straightforward application of the power rule is not sufficient. Techniques such as the product rule, quotient rule, and trigonometric differentiation are essential for dealing with products, quotients, or trigonometric functions, respectively. Moreover, implicit differentiation is used when functions are given in implicit form and not explicitly solvable for one variable in terms of the other.
Implicit differentiation is particularly useful in finding the derivative of inverse functions, without having to express one variable in terms of the other. Yet another technique, logarithmic differentiation, is especially helpful when dealing with products or quotients of functions that are raised to variable powers, similar to our exercise. By employing a combination of these techniques, we can tackle nearly any derivative we confront in calculus, ensuring that we have a robust set of tools for analyzing and understanding how functions change.