Transient voltage of an inductor in an RL circuit
The circuit you’ve shown is an RL circuit with a step  voltage input thus it would give a step response. This step response would be a inductor current that is of the form of a rising exponential like shown below. It’s inductor voltage would be a decaying exponential. The current rises because as we approach steady state the inductor acts like a short. The voltage is at peak initially because the magnet field in the inductor produces a voltage instantly, then voltage decays. 



Source: Fundamentals of electric circuits 5th ed 
Alexander/Sadiku

I believe when you derived the inductor current you took the derivative incorrectly and got a negative value. Only in a source free RL circuit would we get a negative voltage across inductor because of the rapid collapse of magnetic field producing a voltage as to try to maintain current through inductor.