91Ó°ÊÓ

When dealing with logarithms, a useful rule to remember is the Logarithm Power Rule. This rule states that \( \ln a^b = b \ln a \). Essentially, you can take the exponent in a logarithmic expression and place it in front as a coefficient.
This transformation is handy because it simplifies logarithmic functions, making differentiation easier.
In the given exercise, the function \( f(x) = \ln x^2 \) can be rearranged to \( f(x) = 2 \ln x \) using this rule.
The Logan' theorem not only simplifies calculations but also helps to neatly express derivations. Remembering and applying this rule correctly is crucial when you need to differentiate functions that involve powers of variable quantities within a logarithm.

The constant multiple rule is a basic yet powerful tool in differentiation. It says that for any constant \( c \) and function \( g(x) \), the derivative of \( c g(x) \) is \( c \cdot g'(x) \).
In simple terms, if you have a constant multiplied by a function, you can "take the constant out," differentiate the function, and then multiply the result by that constant.
In our exercise, after rewriting \( f(x) = \ln x^2 \) as \( f(x) = 2 \ln x \), you encounter a constant, \( 2 \), multiplied by \( \ln x \).
By applying the constant multiple rule, the derivative of \( 2 \ln x \) becomes \( 2 \cdot \left(\frac{1}{x}\right) = \frac{2}{x} \).
This step is crucial because it allows you to easily handle and simplify the process of calculating derivatives when constants are involved.

Differentiation is what we use to find how a function changes, or more formally, to find the derivative. Several techniques are pivotal to successfully differentiate different types of functions.

• Power Rule: Useful for polynomial functions, where you bring down the exponent as a coefficient and reduce the exponent by one: \( \frac{d}{dx}[x^n] = nx^{n-1} \).


• Logarithmic Differentiation: Applies particularly when you are dealing with logarithmic functions, as seen in our example with \( \ln x \), where the derivative is \( \frac{1}{x} \).

Understanding and applying these techniques are essential in calculus.
They provide the foundational skills necessary to differentiate complex functions efficiently. Using these techniques together, as we did to find \( f'(x) = \frac{2}{x} \), shows how multiple rules can seamlessly work together in solving derivative problems.