How do i prove that a = v * dv/dx?

my book says that a =dv/dt = dv/dx * dx/dt = dv/dx * v

that looks all ok but according to this equation it seems that x(displacement) is a function of t whereas x=v * t. so how can we differentiate x without differentiating v which leads to circular reasoning?

How are the crust and the inner core alike? a) they are both solid. b) they both have the same temperature. c) they are both under the same pressure. d) they are both very close to the center of the earth.

Along wire of diameter 1 mm is submerged in an oil bath of temperature ∞ 31°c. the wire has an electrical resistance per unit length of 0.01 ω/m. if a current of 103 a flows through the wire and the convection coefficient is 500 w/m2·k, what is the steady-state temperature of the wire? from the time the current is applied, how long does it take for the wire to reach a temperature that is within 1°c of the steady-state value? the properties of the wire are 8000 kg/m3, 500 j/kg·k, and 20 w/m·k.

How does an increase in temperature generally affect the rate of a reaction?