How to differentiate implicitly with respect to x?

Question

I don’t understand how you get the 1/y* dy/dx = ln3 from the equation lny = xln3

Answer 1

It's the chain rule. $\frac{d}{dy} \ln(y) = \frac{1}{y}$ and so when $y$ is a function of $x$ we have

$$\frac{d}{dx} ln(y(x))=\frac{1}{y(x)}\frac{dy(x)}{dx}$$

This trick is called logarithmic differentiation. Sometimes, (especially when are trying to maximize since log is an increasing function) we take the natural log on both sides and then take the derivative.