Question

Please find the differenciatial coefficient with respect to x.

Explaination needed ❤​

Answer 1

Given,

$\longrightarrow y=\dfrac{1}{5^x}$

Since $\dfrac{1}{a^n}=a^{-n},$

$\longrightarrow y=5^{-x}$

Since $a^b=e^{b\log a},$

$\longrightarrow y=e^{-x\log 5}$

Differentiating with respect to $x,$

$\longrightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}\left[e^{-x\log 5}\right]$

By chain rule,

$\longrightarrow \dfrac{dy}{dx}=\dfrac{d\left[e^{-x\log 5}\right]}{d[-x\log 5]}\cdot\dfrac{d[-x\log 5]}{dx}$

$\longrightarrow\underline{\underline{\dfrac{dy}{dx}=-e^{-x\log 5}\log 5}}$

Thus differentiated!