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I am new to calculus. Let $f(X)$ be continuous and differentiable over a set of variables $X$ where $x \in X$ and $x \in (0,\infty)$. Is there any general solution or rules to solve $\frac{f'(X)}{f(X)}$?
1 Answer
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Of course if you can differentiate $f(x)$ you can immediately write $\frac{f'(x)}{f(x)}$ by just dividing. For example, if $$ f(x) = x^3 + 5 \\ f'(x) = 3x^2 \\ \frac{f'(x)}{f(x)} = \frac{3x^2}{x^3+5} $$ But I suspect what your professor or teacher is getting at is that it is sometimes easier if you realize that $$ \frac{d(\log(f(x))}{dx} = \frac{f'(x)}{f(x)} $$ For example, if $$ f(x) = e^{\sin x}\\ \frac{f'(x)}{f(x)} = \cos x $$ without much work, since $\log\left( e^{\sin x} \right)= \sin x$.
