Answer:
[tex]\displaystyle\frac{dy}{dx} = \frac{-2}{x\log 2}[/tex]
Step-by-step explanation:
We are given the function:
[tex]y = \log_2\bigg(\dfrac{1}{x^2}\bigg)[/tex]
We have to differentiate the given function.
Formula used:
[tex]\log_a b = \dfrac{\log b}{\log a}\\\\\dfrac{d(\log x)}{dx} = \dfrac{1}{x}\\\\\dfrac{d(x^n)}{dx} = nx^{n-1}[/tex]
The derivation takes place in the following manner:
[tex]\displaystyle\frac{dy}{dx} = \frac{d}{dx} \bigg(\log_2\dfrac{1}{x^2}\bigg)\\\\=\frac{d}{dx}\bigg(\frac{\log(\frac{1}{x^2})}{\log 2}\bigg)\\\\=\frac{1}{\log 2}\frac{d}{dx} \bigg(\log\dfrac{1}{x^2}\bigg)\\\\=\frac{1}{\log 2}\bigg(\frac{1}{\frac{1}{x^2}}\bigg)\frac{d}{dx}(\frac{1}{x^2})\\\\=\frac{x^2}{\log 2}\bigg(\frac{-2}{x^3}\bigg)\\\\=\frac{-2}{x\log 2}[/tex]
