Remember that the derivative of a function defines the slope of the line tangent to it at that point.
If you picture the right triangle formed by ℓ (the hypotenuse) with the x-axis and angle ϕ at their intersection, its slope can be characterized by the ratio of the opposite side over the adjacent side. This is the definition of the tan function.
The derivative of y = 4x3 is dy/dx = 12x2
tanϕ = 12x2, or for x = 1, tanϕ = 12, so ϕ = the inverse tangent of 12.