Derivative of Natural Logarithm to the Seventh Power

Find the derivative of $(\ln x)^7$.

In this problem, we are tasked with finding the derivative of a composite function, specifically, the seventh power of the natural logarithm of x. This requires applying rules of differentiation suited for composite and power functions. The crucial tool for solving this problem is the chain rule, which allows us to differentiate composite functions systematically. 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 itself. In mathematical terms, if you have a function y = f(g(x)), the derivative dy/dx would be $f'(g(x)) \times g'(x)$. This is essential for handling the nested nature of our function $(\ln x)^7$. Another key concept is the power rule for derivatives, which states that the derivative of $x^n$ is $n \times x^{n-1}$. When combined, these rules enable us to tackle more complex functions like the one given. Understanding these foundational tools and when to apply them is integral to solving such calculus-based problems effectively. It showcases how we deconstruct and attack problems involving complex compositions by breaking them down into recognizable parts. This exercise not only reinforces the mechanics of differentiation but also enhances critical thinking about functions and their behavior.

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