poetinjam4gj4

2023-03-26

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

iistiidgllj

Take both sides' natural logarithms.
$\mathrm{ln}\left(f\left(x\right)\right)=\mathrm{ln}\left({5}^{\mathrm{ln}x}\right)$
Using the following rule, the right side can be made simpler: $\mathrm{log}\left({a}^{b}\right)=b\mathrm{log}\left(a\right)$
$\mathrm{ln}\left(f\left(x\right)\right)=\mathrm{ln}\left(x\right)\mathrm{ln}\left(5\right)$
Distinguish the opposing sides. Remember that the left-hand side will employ the chain rule. Remember that $\mathrm{ln}\left(5\right)$ is just a constant and will remain on the right-hand side.
$\frac{1}{f\left(x\right)}\cdot f\prime \left(x\right)=\frac{1}{x}\cdot \mathrm{ln}\left(5\right)$
To solve for f'(x), the derivative, multiply both sides by f(x).
$f\prime \left(x\right)=\frac{\mathrm{ln}\left(5\right)}{x}\cdot f\left(x\right)$
Rewrite f(x) as ${5}^{\mathrm{ln}x}$.
$f\prime \left(x\right)=\frac{{5}^{\mathrm{ln}x}\mathrm{ln}5}{x}$

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