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### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 37: Parametric & vector-valued function differentiation

# Vector-valued functions differentiation

Visualizing the derivative of a position vector valued function. Created by Sal Khan.

## Want to join the conversation?

• So, would the magnitude of the tangent vector essentially be infinite, since as the h approaches zero, the magnitude gets larger? Or am I missing something?
• Remember, as h approaches zero then r(t+h) approaches r(t) such that r(t+h)-r(t) is an infinitesimally small tangent vector. when you divide a very small quantity with another comparable quantity you get a reasonably sized quantity. e.g. 0.0000000000024 / 0.0000000000006 = 4.
So, you won't get an infinitely large tangent vector.
• at , he says "horizontal", doesn't he mean "vertical"?
• Yep, he misspoke. He does name it correctly elsewhere in the video, though.
• What's the difference between taking a gradient and the derivative of a position vector?
• There are several differences. First, the gradient is acting on a scalar field, whereas the derivative is acting on a single vector. Also, with the gradient, you are taking the partial derivative with respect to x, y, and z: the coordinates in the field, while with the position vector, you are taking the derivative with respect to a single parameter, normally t. Finally, the result of a gradient is a vector field while the result of a derivative of a position vector is just another single vector.
• Where is the next video (giving intuition on magnitude) that Sal is talking about?
• Anyone know when and where this normally covered in the academic track?
• Vectors are generally introduced as early as advanced high school mathematics but are not covered in this capacity until Calculus 2 (or equivalent course). They are heavily used in Calculus 3 (or equivalent) as well as Physics.
• Is anyone else concerned about Sal's functions failing the vertical line test?
• Both x and y are functions of a variable t, which isn't plotted. What's plotted is a curve the function makes as t varies in some interval.
• can't we just say that

dr/dt=d/dt(r)=d/dt(x(t)i+y(t)j)=d/dt(x(t)i)+d/dt(y(t)j)=id/dt(x)+jd/dt(y)=dx/dt i +dy/dt j
• Yeah, I thought this whole video was pretty self-explanatory as well. But I guess it shows it more rigorously.
• what does dr/dt actually denotes in the graph mentioned in the video ,which in the case of the other usual graphs denotes the slope of the tangent at that point?