Questions: Example 4 (a) Show that for a general function f(x), we have: f'(x)=f(x)[ln f(x)]' Hint: Apply the Chain Rule to [ln f(x)]' and simplify. This is known as logarithmic differentiation.

Solution Steps

To show that \( f^{\prime}(x) = f(x)[\ln f(x)]^{\prime} \), we can use logarithmic differentiation. The idea is to take the natural logarithm of both sides of the equation \( y = f(x) \), differentiate implicitly with respect to \( x \), and then solve for \( f^{\prime}(x) \).

To find \( [\ln f(x)]^{\prime} \), we apply the chain rule. The derivative of \(\ln f(x)\) with respect to \(x\) is given by: \[ [\ln f(x)]^{\prime} = \frac{f^{\prime}(x)}{f(x)} \]

We want to express \( f^{\prime}(x) \) in terms of \( f(x) \) and \( [\ln f(x)]^{\prime} \). From the expression for \( [\ln f(x)]^{\prime} \), we have: \[ f^{\prime}(x) = f(x) \cdot [\ln f(x)]^{\prime} \]

Final Answer