I just got to a point in multivariable calculus where I realise I can solve problems in assignments and tests but have no actual idea of what I'm doing. So I started thinking about stuff and came up with a few questions: 1. Is picturing the derivative as the slope of the tangent line to a point of the graph just a useful analogy? Like imagining the integral is the area under the curve? 2. What is the meaning of taking the derivative of a vector? I get that you can take the derivative of each component separately with respect to a parameter that they all depend on. But what does that mean? 3. How is taking the derivative of a vector that describes a curve in space related to taking the derivative (or partial derivatives) of an equation that describes the same curve?