Not sure how to partially differentiate with respect to another term

[jokermaos]

Basically, I don't understand how to get from the first line to the second line (particularly on the RHS)?

How does partially differentiating ln(wage) lead to dwage/wage? I need to understand the concept so from first principles please no shortcuts.

Thank you.

 

[HallsofIvy]

First, since this is a question about differentiating, it should be under "Calculus" not "differential equations. Second, the partial derivative of a function of several variables with respect to one of those variables is just the usual derivative treating all other variables as constants.

Second, you don't go from the first equation to the second. The first line is incorrectly written- there should be no $\displaystyle \frac{\partial}{\partial Education}$ because you have not yet done the derivative on the right side of the equation.

What is true is that if you have $\displaystyle ln(Wage(Education, Experience, Age))= \beta_0+ \beta_1Education+ \beta_2Experiece+ \beta3Age$

Then $\displaystyle \frac{\partial}{\partial Education}ln(Wage)$ on the left side of the equation, is on the right, $\displaystyle \frac{1}{Wage}$ times the partial derivative of $\displaystyle \beta_0+ \beta_1Education+ \beta_2Experiece+ \beta3Age$ which, assuming the "betas" are all constants, is just $\displaystyle \beta_1$.
So $\displaystyle \frac{\partial}{\partial Education}ln(Wage(Education, Experience, Age))= \frac{\beta_1}{Wage}$