91Ó°ÊÓ

The chain rule is a fundamental concept in calculus used to differentiate composite functions effectively. Imagine having two functions, where one is nested inside the other—like layers of an onion. If you have a composite function in the form of \(f(g(x))\), the chain rule helps us find its derivative. It states that the derivative of \(f(g(x))\) is found by:

• First, finding the derivative of the outer function, \(f\), evaluated at the inner function \(g(x)\).
• Second, multiplying that result by the derivative of the inner function, \(g(x)\).

For example, in the expression \((\ln x)^{11}\), notice how it's a composition of a logarithmic and a power function. Here, consider \(u = \ln x\) and then \(f(u) = u^{11}\). By using the chain rule, the process becomes straightforward. Differentiate the outer function as if it were \(u^{11}\), then multiply it by the derivative of the inner function, \(\ln x\). This method simplifies complex derivatives into manageable steps.

The natural logarithm, denoted as \(\ln x\), is a logarithmic function with a base of \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. This function is the inverse of the exponential function \(e^x\) and has specific properties that make it unique:

• The derivative of \(\ln x\) is \(\frac{1}{x}\), which is fundamental when applying the chain rule for composite functions involving logarithms.
• Its domain includes all positive real numbers, meaning \(x > 0\).

In differentiation, the natural logarithm's derivative is crucial when combined with another function, as seen in expressions like \((\ln x)^{11}\). Here, its derivative provides the inner function's rate of change, allowing us to apply the chain rule efficiently. Understanding the natural logarithm and its properties simplify the process of solving calculus problems that involve logarithmic differentiations.

The power rule is an essential tool in calculus for finding the derivative of polynomial and power functions quickly. It states that if you have a function of the form \(x^n\), where \(n\) is any real number, the derivative is determined as follows:

• Bring down the exponent \(n\) in front as a coefficient.
• Decrease the exponent by one.

For example, if \(f(x) = x^{11}\), the derivative \(f'(x)\) would be \(11x^{10}\). This straightforward method is vital for finding the rate of change of functions raised to a power.
In our specific exercise, we applied the power rule to differentiate \((u = \ln x)^{11}\). By considering it as \(u^{11}\), the derivative happens seamlessly to become \(11u^{10}\). The power rule simplifies problems where you deal with functions raised to significant powers, working hand-in-hand with the chain rule for composite functions.