fix typo

preliminaries/index.md

@@ -1280,7 +1280,7 @@ slides: true
             <img src="00.Preliminaries.key-stage-0094anim0.png" data-images="00.Preliminaries.key-stage-0094anim0.png,00.Preliminaries.key-stage-0094anim1.png,00.Preliminaries.key-stage-0094anim2.png,00.Preliminaries.key-stage-0094anim3.png,00.Preliminaries.key-stage-0094anim4.png,00.Preliminaries.key-stage-0094anim5.png" class="slide-image" />
 
             <figcaption>
-            <p    >Here's a visualization of what it means to take the partial derivative as a point <span class="red">x</span>=0, <span class="blue">y</span>=5,<span class="red"> </span><em class="red">with respect to x</em>.<br></p><p    >We let the function vary with <span class="red">x</span>, keeping <span class="blue">y</span> fixed. This gives us a kind of "slice" through the surface f. The result is that we have a one-dimensional function again, for which we can take the derivative. We do this by only caring about the way f changes if we change <span class="red">x</span>. In other words, by treating <span class="blue">y</span> as a constant.<br></p><p    >Then we do the same for <span class="blue">y</span>, treating <span class="red">x</span> as a constant.<br></p><p    >For both curves, the derivatives give us the slope of the tangent line. We've shown these as dotted lines here.<br></p><p    >These lines cross the same point, so together, they lie in a shared <em>plane</em>. In higeher dimensions, the tangent lines of all partial derivatives lie in a shared <em>hyperplane</em>. This is the <em>tangent hyperplane</em>. The hyperplane that <em>just</em> touches the surface of f. <br></p><p    >How do we describe the tangent hyperplane? Note what the tangent slopes indicate: how much the plane moves up if we take a step of 1 along the <span class="red">x</span> axis and how much the plane moves up if we take a step along the <span class="red">y</span> axis. These are exactly the roles of the constants in the function <span class="red">x</span>w<sub>1</sub> + <span class="blue">y</span>w<sub>2</sub> + b. Or, in vector notation the elements of the vector w in the function <strong>x</strong><sup>T</sup><strong>w</strong> +<span class="blue"> b</span>.<br></p><p    >This tells us that if we take all of our partial derivatives and stick them in a vector <strong>w</strong>, the function that will describe our tangent hyperplane is <strong>x</strong><sup>T</sup><strong>w</strong> + <span class="blue">b</span> (for some value of <span class="blue">b</span>, which we don't usually care about).<br></p><p    ></p>
+            <p    >Here's a visualization of what it means to take the partial derivative as a point <span class="red">x</span>=0, <span class="blue">y</span>=5,<span class="red"> </span><em class="red">with respect to x</em>.<br></p><p    >We let the function vary with <span class="red">x</span>, keeping <span class="blue">y</span> fixed. This gives us a kind of "slice" through the surface f. The result is that we have a one-dimensional function again, for which we can take the derivative. We do this by only caring about the way f changes if we change <span class="red">x</span>. In other words, by treating <span class="blue">y</span> as a constant.<br></p><p    >Then we do the same for <span class="blue">y</span>, treating <span class="red">x</span> as a constant.<br></p><p    >For both curves, the derivatives give us the slope of the tangent line. We've shown these as dotted lines here.<br></p><p    >These lines cross the same point, so together, they lie in a shared <em>plane</em>. In higher dimensions, the tangent lines of all partial derivatives lie in a shared <em>hyperplane</em>. This is the <em>tangent hyperplane</em>. The hyperplane that <em>just</em> touches the surface of f. <br></p><p    >How do we describe the tangent hyperplane? Note what the tangent slopes indicate: how much the plane moves up if we take a step of 1 along the <span class="red">x</span> axis and how much the plane moves up if we take a step along the <span class="red">y</span> axis. These are exactly the roles of the constants in the function <span class="red">x</span>w<sub>1</sub> + <span class="blue">y</span>w<sub>2</sub> + b. Or, in vector notation the elements of the vector w in the function <strong>x</strong><sup>T</sup><strong>w</strong> +<span class="blue"> b</span>.<br></p><p    >This tells us that if we take all of our partial derivatives and stick them in a vector <strong>w</strong>, the function that will describe our tangent hyperplane is <strong>x</strong><sup>T</sup><strong>w</strong> + <span class="blue">b</span> (for some value of <span class="blue">b</span>, which we don't usually care about).<br></p><p    ></p>
             </figcaption>
             <span class="hint">click image for animation</span>
        </section>