Question regarding the tangent line of a partial derivative and its graph?

[watchintv]

So I am following the video located here: https://www.khanacademy.org/math/mu...-derivatives/v/partial-derivatives-and-graphs

Given the following multi-variable equation f(x,y) = x^2(y) + sin(y)

The derivative with respect to x = 2xy

And, the derivative with respect to y = x^2 + cos(y)

My question is after I figure out the instantaneous slope at a certain point, lets say (1,-1) as in the video. When evaluating the derivative with respect to x at the point 1,-1, you get -2.

So with that, how do I plot a tangent line that passes through that point on a graph with that specific slope (-2)? And am I correct in thinking that there are *two* tangent lines in a multi-variable function, one for x and one for y?

 

[Dr.Peterson]

So I am following the video located here: https://www.khanacademy.org/math/mu...-derivatives/v/partial-derivatives-and-graphs

Given the following multi-variable equation f(x,y) = x^2(y) + sin(y)

The derivative with respect to x = 2xy

And, the derivative with respect to y = x^2 + cos(y)

My question is after I figure out the instantaneous slope at a certain point, lets say (1,-1) as in the video. When evaluating the derivative with respect to x at the point 1,-1, you get -2.

So with that, how do I plot a tangent line that passes through that point on a graph with that specific slope (-2)? And am I correct in thinking that there are *two* tangent lines in a multi-variable function, one for x and one for y?

I haven't watched the video, and won't until you show that you are still interested after the unfortunate delay in being noticed (and tell us what part to watch).

But since this is a surface, you should know that there is not a single tangent line with a single slope, but a tangent plane. Does the video not say that?

 

[Dr.Peterson]

So with that, how do I plot a tangent line that passes through that point on a graph with that specific slope (-2)? And am I correct in thinking that there are *two* tangent lines in a multi-variable function, one for x and one for y?

I went ahead and watched the video, since it's not long. He shows two tangent lines, one in the plane y=1:

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and another in the plane x=-1:

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If you cut the surface with any other plane, you could make a tangent line in that plane. So there are infinitely many.

All these tangent lines together constitute the tangent plane. The two particular tangent lines he plotted are associated with the two partial derivatives he calculated.

What was your specific question, beyond what he explained? If you're asking how he plotted it, he clearly used some software; it's not easy to do this by hand. But you could find another point on the tangent line, plot that point, and draw a line through both points. You have to imagine the line being in the appropriate plane, as he showed.