fix typo in 'derivatoves' lecture 0 slide 81

preliminaries/index.md

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             <img src="00.Preliminaries.key-stage-0081anim0.svg" data-images="00.Preliminaries.key-stage-0081anim0.svg,00.Preliminaries.key-stage-0081anim1.svg" class="slide-image" />
 
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-            <p    >Here are two forms of shorthand we will occasionally allow ourselves, to simplify the notation. At the top, if it is clear that the result of <span>f</span> is a function of <span class="red">x</span>, we will omit its argument. You can think of this in Leibniz's terms if you like: we divide the change in f by the change in x. What is left more implicit by this notation is that we create the change in x, and observe the resulting change in <span>f</span>.<br></p><aside    >In other words, we think of <span>f</span> as referring both to the function, <span>and</span> a variable representing its output. This is a little ambiguous, but if it's clear from context what we mean, it simplifies things a lot.<br></aside><p    >The second line shows what happens when we fill in the explicit functional form of <span>f</span> in terms of <span class="red">x</span>. We will do this very often. Technically, we need to put brackets around the whole function, or the statement might be ambiguous (especially if we write the function to the right of the division line as in the previous slide). In practice, things look a lot clearer without the brackets, so if the potential ambiguity is minimal, or can easily be resolved from context, we allow ourselves to leave the brackets out.</p><p    ></p>
+            <p    >Here are some examples. The first two lines show the derivatives we've already worked out in the Lagrange notation.<br></p><p    >The right part of the second line shows how much clearer things can become when we assume that we know which variable is dependent on which. The notation is more ambiguous, but a lot clearer.<br></p><p    >The third line shows the benefit of indicating <span class="red">the independent variable</span>. The variables a, b, c are indicated with letters in the function, but we treat them as constants: x is the only variable we change to observe the resulting change in the function above the line. The rest is treated the same way as the 3 in the exponent is. <br></p><aside    >This can get a little confusing when we combine it with the shorthand from the second line. We assume that the reader knows whether <span>a</span> is a constant, or a function of <span>a(x)</span>. If things get too ambiguous, we can always spell them out more explicitly.</aside><aside    ></aside>
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             <span class="hint">click image for animation</span>
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