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<title>Partial Derivatives</title>
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<h1 class="center">Partial Derivatives</h1>
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<p> </p>
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<p>A Partial Derivative is a <a href="derivatives-introduction.html">derivative</a> where we hold some variables constant. Like in this example:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/partial-derivative.gif" alt="partial derivative on surface" height="339" width="300"></p>
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<h3>Example: a function for a surface that depends on two variables <b>x</b> and <b>y</b></h3>
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<p> </p>
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<p>When we find the slope in the <b>x</b> direction (while keeping <b>y</b> fixed) we have found a partial derivative.</p>
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<p> </p>
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<p>Or we can find the slope in the <b>y</b> direction (while keeping <b>x</b> fixed).</p>
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</div>
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<p> </p>
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<p>Let's first think about a function of <b>one variable</b> (x):</p>
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<p class="center large">f(x) = x<sup>2</sup></p>
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<p>We can find its <a href="derivatives-rules.html">derivative</a> using the <a href="power-rule.html">Power Rule</a>:</p>
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<p class="center large">f’(x) = 2x</p>
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<p class="larger">But what about a function of <b>two variables</b> (x and y):</p>
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<p class="center large">f(x, y) = x<sup>2</sup> + y<sup>3</sup></p>
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<p>We can find its <b>partial</b> derivative <b>with respect to x</b> when we treat <b>y as a constant</b> (imagine y is a number like 7 or something):</p>
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<p class="center large">f’<sub>x</sub> = 2x + 0 = 2x</p>
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<div class="center80">
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<p><i>Explanation:</i></p>
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<ul>
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<li><i>the derivative of x<sup>2</sup> (with respect to x) is 2x</i></li>
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<li><i> we <b>treat y as a constant</b>, so y<sup>3</sup> is also a constant (imagine y=7, then 7<sup>3</sup>=343 is also a constant), and the derivative of a constant is 0</i></li>
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</ul>
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</div>
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<p>To find the partial derivative <b>with respect to y</b>, we treat <b>x as a constant</b>:</p>
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<p class="center large">f’<sub>y</sub> = 0 + 3y<sup>2</sup> = 3y<sup>2</sup></p>
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<div class="center80">
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<p><i>Explanation:</i></p>
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<ul>
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<li><i>we now <b>treat x as a constant</b>, so x<sup>2</sup> is also a constant, and the derivative of a constant is 0</i></li>
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<li><i>the derivative of y<sup>3</sup> (with respect to y) is 3y<sup>2</sup></i></li>
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</ul>
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</div>
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<p> </p>
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<p>That is all there is to it. Just remember to treat <b>all other variables as if they are constants</b>.</p>
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<p> </p>
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<h3>Holding A Variable Constant</h3>
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<p>So what does "holding a variable constant" look like?</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/cylinder-dimensions.svg" alt="Cylinder Dimensions"></p>
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<h3>Example: the volume of a cylinder is V = <span class="times">π</span> r<sup>2</sup> h</h3>
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<p>We can write that in "multi variable" form as</p>
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<p class="center large">f(r, h) = <span class="times">π</span> r<sup>2</sup> h</p>
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<p> </p>
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<p>For the partial derivative with respect to r we hold <b>h constant</b>, and r changes:</p>
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<div style="clear:both"></div>
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<p class="center"><img src="images/cylinder-r-changes.svg" alt="Cylinder with r changing"></p>
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<p class="center large">f’<sub>r</sub> = <span class="times">π</span> (2r) h = 2<span class="times">π</span>rh</p>
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<p class="center"><i>(The derivative of r<sup>2</sup> with respect to r is 2r, and <span class="times">π</span> and h are constants)</i></p>
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<p>It says "as only the radius changes (by the tiniest amount), the volume changes by 2<span class="times">π</span>rh"</p>
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<p>It is like we add a skin with a circle's circumference (2<span class="times">π</span>r) and a height of h.</p>
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<p> </p>
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<p>For the partial derivative with respect to h we hold <b>r constant</b>:</p>
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<p class="center"><img src="images/cylinder-h-changes.svg" alt="Cylinder with r changing"></p>
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<p class="center large">f’<sub>h</sub> = <span class="times"> π</span> r<sup>2 </sup>(1)= <span class="times"> π</span>r<sup>2</sup></p>
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<p class="center"><i>(<span class="times">π</span> and r<sup>2</sup> are constants, and the derivative of h with respect to h is 1)</i></p>
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<p>It says "as only the height changes (by the tiniest amount), the volume changes by <span class="times">π</span>r<sup>2</sup>"</p>
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<p>It is like we add the thinnest disk on top with a circle's area of <span class="times">π</span>r<sup>2</sup>.</p>
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</div>
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<p>Let's see another example.</p>
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<div class="example">
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<h3>Example: The surface area of a square prism.</h3>
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<p class="center"><img src="images/partial-derivative-box.svg" alt="Cylinder with r changing"></p>
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<p>The surface includes the top and bottom with areas of <b>x<sup>2</sup></b> each, and 4 sides of area <b>xy</b> each:</p>
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<p class="center large">f(x, y) = 2x<sup>2</sup> + 4xy</p>
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<p class="so">f<span class="center large">’<sub>x</sub></span> = 4x + 4y</p>
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<p class="so">f<span class="center large">’<sub>y</sub></span> = 0 + 4x = 4x</p>
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</div>
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<h3>Three or More Variables</h3>
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<p>We can have 3 or more variables. Just find the partial derivative of each variable in turn while treating<b> all other variables as constants</b>.</p>
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<div class="example">
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<h3>Example: The volume of a cube with a square prism cut out from it.</h3>
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<p class="center"><img src="images/partial-derivative-box-cube.svg" alt="Cylinder with r changing"></p>
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<p class="center large">f(x, y, z) = z<sup>3</sup> − x<sup>2</sup>y</p>
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<p class="so">f’<sub>x</sub> = 0 − 2xy = −2xy</p>
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<p class="so">f’<sub>y</sub> = 0 − x<sup>2</sup> = −x<sup>2</sup></p>
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<p class="so">f’<sub>z</sub> = 3z<sup>2</sup> − 0 = 3z<sup>2</sup></p>
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</div>
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<p>When there are many x's and y's it can get confusing, so a mental trick is to change the "constant" variables into letters like "c" or "k" that <i>look</i> like constants.</p>
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<div class="example">
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<h3>Example: f(x, y) = y<sup>3</sup>sin(x) + x<sup>2</sup>tan(y)</h3>
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<p>It has x's and y's all over the place! So let us try the letter change trick.</p>
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<p>With respect to x we can change "y" to "k":</p>
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<p>f(x, y) = <span class="hilite">k</span><sup>3</sup>sin(x) + x<sup>2</sup>tan(<span class="hilite">k</span>)</p>
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<p class="so">f’<sub>x</sub> = k<sup>3</sup>cos(x) + 2x tan(k)</p>
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<p>But remember to turn it back again!</p>
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<p class="so">f’<sub>x</sub> = y<sup>3</sup>cos(x) + 2x tan(y)</p>
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<p>Likewise with respect to y we turn the "x" into a "k":</p>
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<p>f(x, y) = y<sup>3</sup>sin(<span class="hilite">k</span>) + <span class="hilite">k</span><sup>2</sup>tan(y)</p>
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<p class="so">f’<sub>y</sub> = 3y<sup>2</sup>sin(k) + k<sup>2</sup>sec<sup>2</sup>(y)</p>
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<p class="so">f’<sub>y</sub> = 3y<sup>2</sup>sin(x) + x<sup>2</sup>sec<sup>2</sup>(y)</p>
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<p>But only do this if you have trouble remembering, as it is a little extra work.</p>
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</div>
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<p> </p>
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<div class="words">
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<p><b>Notation</b>: we have used <b>f’<sub>x</sub></b> to mean "the partial derivative with respect to x", but another very common notation is to use a funny backwards d (∂) like this:</p>
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<p class="center large"><span class="intbl"><em>∂f</em><strong>∂x</strong></span> = 2x</p>
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<p>Which is the same as:</p>
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<p class="center large">f’<sub>x</sub> = 2x</p>
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<p><span class="larger">∂</span> is called "del" or "dee" or "curly dee"</p>
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<p>So <span class="intbl">
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<em>∂f</em>
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<strong>∂x</strong></span> can be said "del f del x"</p>
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</div>
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<div class="example">
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<h3>Example: find the partial derivatives of <b>f(x, y, z) = x<sup>4</sup> − 3xyz</b> using "curly dee" notation</h3>
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<p>f(x, y, z) = x<sup>4</sup> − 3xyz</p>
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<p class="so"><span class="intbl"><em>∂f</em><strong>∂x</strong></span> = 4x<sup>3</sup> − 3yz</p>
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<p class="so"><span class="intbl"><em>∂f</em><strong>∂y</strong></span> = −3xz</p>
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<p class="so"><span class="intbl"><em>∂f</em><strong>∂z</strong></span> = −3xy</p>
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</div>
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<p>You might prefer that notation, it certainly looks cool.</p>
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<p> </p>
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<div class="questions">
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<script>getQ(13373, 13374, 13375, 13376, 13377, 13378, 13379, 13380, 13381, 13382, 13383);</script>
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<a href="index.html">Calculus Index</a>
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