Find the derivative of a function ln ⁡ x x .

The quotient rule states that:
$(\frac{f}{g})=\frac{f\prime g-fg\prime }{g^2}$
where the apostrophe $(\prime )$ means "the derivative of".
In this example, we let $f=\ln x$ and $g=x$
This allows us to rewrite the function as:
$\frac{((\frac{d}{dx}\ln x)x-\ln x((\frac{d}{dx})x)}{x^2}$
Other calculus rules tell us that the derivative of $\ln x$ is always $\frac{1}{x}$. As a result, the first half of our numerator becomes $x(\frac{1}{x})$, Which neatly simplifies to $1$. This gives:
$\frac{1-\ln x((\frac{d}{dx})x)}{x^2}$
Now we can use the power rule, which states that $(\frac{d}{dx})x^n=n{x}^{n-1}$. Since we know that nn in this case is $1$ (because $\mathrm{x}$ has no exponent), this becomes $1\times x^{1-1}$, which yields a value of $1$. This gives us our final answer:
$\frac{1-\ln x}{x^2}$