Tyrell Singleton

2023-02-26

How to find the derivative of $y=\mathrm{ln}\left(5x\right)$?

Kylie Woodward

Suppose, $y=\mathrm{ln}\left(b\left(x\right)\right)$
Thus using Chain Rule,
$y\prime =\frac{1}{b\left(x\right)}\cdot \left(b\left(x\right)\right)\prime$
Similarly following for the above function yields,
$y\prime =\frac{1}{5x}\cdot \left(5x\right)\prime$
$y\prime =\frac{1}{5x}\cdot 5$
Hence, $y\prime =\frac{1}{x}$

Rodney Hoover

A student comfortable with the natural logarithm function and its properties might think of this:
One could reason as follows:
$y=\mathrm{ln}\left(5x\right)=\mathrm{ln}\left(5\right)+\mathrm{ln}\left(x\right)$.
But ln(5) is a constant, so its derivative is 0.
Hence, $\frac{dy}{dx}=\frac{d}{dx}\left(\mathrm{ln}5+\mathrm{ln}x\right)=\frac{d}{dx}\left(\mathrm{ln}x\right)=\frac{1}{x}$.

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