Gradients and Partial Derivatives

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<div id="merch-shelf" class="style-scope ytd-watch-flexy">00:03
Suppose we have a function of two variables.
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We have the variable X.
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We have the variable Y.
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And we have Z, which is a function of both X and Y.
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Suppose that we pick one value for X,
00:39
and we keep X at this one value as we change the value for Y.
00:48
Each time we change Y by a small amount, Z also changes by a small amount.
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At each point,
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the change in Z divided by the change in Y is given by the slope of this line.
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Red indicates a negative slope.
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Let’s now pick a different value for X, and keep X at this new value.
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Again, at each point,
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the change in Z divided by the change Y is given by the slope of this line.
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The change in Z divided by the change in Y is what we
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refer to as the partial derivative of Z with respect to Y.
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Every point on the graph has a value for
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the partial derivative of Z with respect to Y.
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Here, green indicates a positive value, and red indicates a negative value.
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Every point on the graph also has a value for
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the partial derivative of Z with respect to X.
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In this case, the value for Y is held constant, while the value for X changes.
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Suppose we represent the slope of this line with an arrow.
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The length and direction of the arrow indicates
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if the slope is positive or negative, and by how much.
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Every point has an arrow that represents
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the partial derivative of Z with respect to X.
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Every point also has an arrow that represents
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the partial derivative of Z with respect to Y.
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If we add these two arrows together as vectors, then we will get a new arrow.
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Let us call this new arrow the gradient of the function Z.
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This arrow always indicates
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the direction in which Z is increasing by the largest amount.
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The length of this arrow indicate by how much Z is increasing in this direction.
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Much more information is available in the other videos on this channel.
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Gradients and Partial Derivatives

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