/sci/ - Science & Math

Could someone point to a resource where I can understand why for example taking a derivative of lnx=1/x, but of lny=1/y*y'? Also would like to understand taking a derivative of something with respect to something. I'm 100% those two concepts are connected, but my "teacher" never explained any. We just learned the rules. And I'd like to understand these through usage of Leibniz notation.

Also in pic related is the second one a second derivative and why is it written as such? I've only ever learned y', y", y"' etc. in school and this is new to me, but the teacher randomly started using d/dx, dt/dy/dx/dt and similar shit without even explaining the concept behind it. Fuck I hate my school.