If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### 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?
• This is a good question, but it's actually not possible to see dr/dt in the graph here, because we only see r(t) graphed in terms of x(t) and y(t) and not directly in terms of t. If we were instead to graph r(t) on the y-axis and t on the x-axis, we would then be able to visualize dr/dt as the slope of the tangent at a given point t. (However, we then couldn't easily visualize the relationship between x(t) and y(t), which this graph allows us to do.)

Conceptually, dr/dt doesn't just mean the slope; it means the instantaneous rate of change of r(t). So for instance, if r(t) is the position of something with respect to time (t), dr/dt would tell us its velocity at any given moment.