91Ó°ÊÓ

Logarithmic functions are the inverses of exponential functions. In simple terms, if you have an exponential function like \(y = b^x\), the logarithmic function would be \(x = \log_b(y)\). Logarithmic functions, especially the natural logarithm \(\ln(x)\), are essential in calculus and many real-world applications because they help reduce multiplication to addition, making complex calculations simpler.
For example, the function given in our problem, \(y = \ln(t^2)\), is a logarithmic function where the variable is \(t\). The natural logarithm, represented by \(\ln\), assumes the base of the logarithm to be \(e\), a mathematical constant approximately equal to 2.718. The use of natural logarithms is common because they have properties that simplify differentiation and integration calculations greatly.
Here are some characteristics of logarithmic functions that help in our calculations:

• \(\ln(1) = 0\), because raising \(e\) to the power of zero gives 1.
• The logarithm of a product is the sum of the logarithms: \(\ln(ab) = \ln(a) + \ln(b)\).
• The logarithm of a power is the exponent times the logarithm of the base: \(\ln(a^b) = b\ln(a)\).

These properties simplify complex functions into more manageable forms, thus making the differentiation process easier.

The chain rule is a fundamental theorem in calculus used to compute the derivative of a composite function. A composite function is essentially a function within another function, like \(f(g(x))\).
When applying the chain rule, we differentiate the outer function first and then multiply it by the derivative of the inner function. The formal expression is: \[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]
For the exercise at hand, our function simplifies using logarithmic properties: \(y = 2\ln(t)\). The chain rule becomes especially handy here, as it allows us to separate the differentiation of the constant multiplied by the logarithm. As per our solution, after applying the properties of logarithms to simplify \(\ln(t^2)\) to \(2\ln(t)\), we need to find the derivative of \(2\ln(t)\).
Using the chain rule, with \(u = \ln(t)\), we find the derivative, knowing the derivative of \(\ln(t)\) is \(\frac{1}{t}\). Thus, the derivative becomes \(2 \cdot \frac{1}{t}\), leading us to the final derivative \( \frac{2}{t} \). The chain rule is powerful and widely used in calculus, as it breaks complex differentiation tasks into simpler ones.

Differentiation of logarithmic functions, especially the natural logarithm \(\ln(x)\), is a fundamental concept in calculus. It simplifies the process of finding rates of change in functions involving logarithms.
The derivative of the natural logarithm \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\). This result is profound because it directly simplifies the process of finding derivatives of functions that are composed of logarithmic expressions.
In our problem, we start with \(y = \ln(t^2)\) but simplify to a form easier to differentiate: \(y = 2\ln(t)\). When differentiating \(2\ln(t)\), we utilized the property of constants in differentiation, where the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function:
\[ \frac{d}{dt}[2 \cdot \ln(t)] = 2 \cdot \frac{1}{t} = \frac{2}{t} \]
This derivative tells us the rate of change of the logarithmic function in terms of \(t\). Knowing how to differentiate \(\ln(x)\) is crucial for students as it appears frequently in calculus tasks involving exponential growth, complex equations, and integration.