How do you differentiate #y=ln(1/x)#?

To differentiate ( y = \ln\left(\frac{1}{x}\right) ), you would use the chain rule of differentiation. The chain rule states that if you have a composite function, you differentiate the outer function first and then multiply by the derivative of the inner function. In this case, the outer function is the natural logarithm (( \ln )), and the inner function is ( \frac{1}{x} ).

The derivative of ( \ln(u) ) with respect to ( u ) is ( \frac{1}{u} ). So, applying the chain rule, the derivative of ( \ln\left(\frac{1}{x}\right) ) with respect to ( x ) would be:

[ \frac{d}{dx} \ln\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} \cdot \frac{d}{dx} \left(\frac{1}{x}\right) ]

[ = \frac{1}{\frac{1}{x}} \cdot (-1) \cdot \frac{1}{{x}^2} ]

[ = -\frac{x}{1} \cdot \frac{1}{{x}^2} ]

[ = -\frac{1}{x} ]

So, the derivative of ( y = \ln\left(\frac{1}{x}\right) ) with respect to ( x ) is ( -\frac{1}{x} ).