91Ó°ÊÓ

The chain rule is a fundamental tool in calculus. It helps us differentiate compositions of functions. Think of it as peeling an onion, layer by layer.

For instance, if we have a function like \(g(h(x))\), where one function \(g\) is inside another function \(h\), the chain rule tells us to differentiate the outer function first and then the inner function.

This is formally written as:
\[ \frac {d}{dx} [g(h(x))] = g'(h(x)) \bullet h'(x) \]

When we applied the chain rule in the given problem, we needed to differentiate the function involving a natural logarithm. Natural logs often simplify differentiation because we can leverage their properties.

The natural logarithm, denoted as \ln(x)\, is a logarithm to the base \e\, where \e\ is approximately 2.71828. Natural logs stand out due to their unique properties and frequent appearance in calculus problems.

A few key properties of natural logarithms are:

• \ln(1) = 0\
• \ln(ab) = \ln(a) + \ln(b)\
• \ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)\
• \ln(a^b) = b \ln(a)\

In our problem, we used the property \[ \ln(3^{\frac{1}{x}}) = \frac{1}{x} \ln(3) \]

This allowed us to turn an exponential function into a more manageable linear expression. Such transformations make differentiation easier.

Differentiation is the process of finding the derivative of a function. It measures how a function changes as its input changes. The derivative of a function \(f(x)\) is denoted as \(f'(x)\) or \frac{df}{dx}\.

There are several rules of differentiation, including

• The power rule: \frac{d}{dx}[x^n] = nx^{n-1}\
• Product rule: \frac{d}{dx}[uv] = u'v + uv'\
• Quotient rule: \frac{d}{dx}\left( \frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\
• Chain rule (discussed above)

In our exercise, after applying the natural logarithm and simplifying, we differentiated using chain rule principles. On the left side, differentiation transforms \ln(f(x))\ into \( \frac{f'(x)}{f(x)} \). The right side simplifies straightforwardly into \( -\frac{\ln(3)}{x^2} \). Hence, the final derivative was \[ f'(x) = -\frac{\ln(3) \bullet 3^{\frac{1}{x}}}{x^2} \]
This combination of steps is what we call logarithmic differentiation.