Step-by-step explanation:
[tex]f(x) = ln( \frac{1}{x} ) [/tex]
To find the derivative, notice we have a function 1/x inside of another function, ln(x)
We use what we call the chain rule,
It states that
derivative of a '
[tex] \frac{d}{dx} f(g(x)) = f '(g(x)) \times \: g '(x)[/tex]
Here f is ln(x)
f is 1/x
So first, we know that
[tex] \frac{d}{dx} ( ln(x) = \frac{1}{x} [/tex]
so
[tex]f'(g(x)) = \frac{1}{ \frac{1}{x} } [/tex]
We know that
[tex]g'(x) = - \frac{1}{ {x}^{2} } [/tex]
So we have
[tex]x \times - \frac{1}{ {x}^{2} } = \frac{ - 1}{x} [/tex]
