 William Burnett

2021-12-30

What is the derivative of ${5}^{x}$ ? Laura Worden

Explanation:
Given: $\frac{d}{dx}\left({5}^{x}\right)$
Apply the general formula that
$\therefore \frac{d}{dx}\left({5}^{x}\right)={5}^{x}\mathrm{ln}5$ Maria Lopez

Explanation:
Let $y={5}^{x}$
differentiate implicity with respect to x
take $\mathrm{ln}$ (natural log) of both sides
$\mathrm{ln}y={\mathrm{ln}5}^{x}=x\mathrm{ln}5$
$⇒\frac{1}{y}\frac{dy}{dx}=\mathrm{ln}5$
$⇒\frac{dy}{dx}=y\mathrm{ln}5={5}^{x}\mathrm{ln}5$ nick1337

For a generalized answer to this question, you can use the following which works in cases of ${5}^{x}$ and ${x}^{5}$:
$\frac{d}{dx}\left(f\left(x{\right)}^{g\left(x\right)}\right)=f\left(x{\right)}^{g\left(x\right)-1}\left(g\left(x\right){f}^{\prime }\left(x\right)+f\left(x\right)\mathrm{log}\left(f\left(x\right)\right){g}^{\prime }\left(x\right)$
So
$\frac{d}{dx}{x}^{5}={x}^{5-1}\left(5×1+x\mathrm{log}\left(x\right)×0\right)$
$={x}^{4}\left(5+0\right)$
$=5{x}^{4}$
And, likewise:
$\frac{d}{dx}{5}^{x}={5}^{x-1}\left(x×0+5\mathrm{log}\left(5\right)×\right)$
$={5}^{x-1}\left(0+5\mathrm{log}\left(5\right)\right)$
$={5}^{x}\mathrm{log}\left(5\right)$
Note $5×{5}^{x-1}\equiv {5}^{x}$

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