liachta6VW

2022-11-25

Find the derivative of a function $\frac{\mathrm{ln}x}{x}$.

Aleah Rowe

Expert

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

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