Why does the video use $h*v$ instead of just v when defining the directional derivative?

Q: Why does the video use h*v instead of just v when defining the directional derivative?

The video uses h*v because the directional derivative is defined as an infinitesimal rate of change. Using the full vector v would represent a finite step, whereas scaling it by a small scalar h and taking the limit as h approaches 0 captures the instantaneous slope in that direction.

The video uses

h*v

because the directional derivative is defined as an infinitesimal rate of change. Using the full vector v would represent a finite step, whereas scaling it by a small scalar h and taking the limit as h approaches 0 captures the instantaneous slope in that direction.

Conditions

The function is differentiable at the point.
The direction is specified by a vector v.
h is a scalar approaching 0.

Reasoning, step by step

Identify the direction vector v.
Scale v by a small scalar h to create an infinitesimal step $h*v$ .
Take the limit as h approaches 0.
Calculate the ratio of the output change to the input displacement h.

Example

For $v=[-1,2]$ , the actual step considered is [-h, 2h] where h is very small, not the full vector [-1,2].

Common misconceptions

Confusing the direction vector with the actual infinitesimal displacement.
Thinking that the directional derivative means literally moving by the full vector v.

Watch the explanation

YouTubeDirectional derivative

2:02 – 3:00Watch this moment ↗

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.

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