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<h1 class="center">Differentiable</h1>
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<p>Differentiable means that the <a href="derivatives-introduction.html">derivative</a> <b>exists</b> ...</p>
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<div class="example">
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<h3>Example: is x<sup>2</sup> + 6x differentiable?</h3>
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<p><a href="derivatives-rules.html">Derivative rules</a> tell us the derivative of x<sup>2</sup> is 2x and the derivative of x is 1, so:</p>
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<p>Its derivative is <b>2x + 6</b></p>
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<p>So yes! x<sup>2</sup> + 6x is differentiable.</p>
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</div>
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<p><b>... and</b> it must exist for <b>every</b> value in the function's <a href="../sets/domain-range-codomain.html">domain</a>.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>
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<h3>Domain</h3>
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<p>In its simplest form the domain is<br>
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all the values that go into a function</p></td>
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<td><img src="../sets/images/range-domain-graph.svg" alt="domain and range" width="298" height="167" ></td>
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</tr>
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</tbody></table>
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<div class="example">
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<h3>Example (continued)</h3>
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<p>When not stated we assume that the domain is the <a href="../numbers/real-numbers.html">Real Numbers</a>.</p>
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<p>For <b>x<sup>2</sup> + 6x</b>, its derivative of <b>2x + 6</b> exists for all Real Numbers.</p>
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<p>So we are still safe: x<sup>2</sup> + 6x is differentiable.</p>
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</div>
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<p>But what about this:</p>
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<div class="example">
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<h3>Example: The function f(x) = |x| (<a href="../sets/function-absolute-value.html">absolute value</a>):</h3>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><b>|x|</b> looks like this:</td>
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<td> </td>
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<td><img src="../sets/images/function-absolute.svg" alt="Absolute Value function" width="241" height="241" ></td>
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</tr>
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</tbody></table>
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<p>At <b>x=0</b> it has a very pointy change!</p>
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<p><b>Does the derivative exist at x=0?</b></p>
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</div>
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<h2>Testing</h2>
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<p>We can test any value "a" by finding if the <a href="limits.html">limit</a> exists:</p>
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<div></div>
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<p class="center large"><span class="intbl"><span class="lim">lim</span>
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<strong>h→0</strong>
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</span> <span class="intbl">
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<em>f(a+h) − f(a)</em>
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<strong>h</strong>
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</span></p>
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<div class="example">
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<h3>Example (continued)</h3>
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<p>Let's calculate the limit for |x| at the value 0:</p>
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<p> </p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="intbl"><span class="lim">lim</span>
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<strong>h→0</strong>
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</span> <span class="intbl">
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<em>f(a+h) − f(a)</em>
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<strong>h</strong>
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</span></span></div>
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<div class="row"><span class="left">f(x) = |x|:</span><span class="right"><span class="intbl"><span class="lim">lim</span>
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<strong>h→0</strong>
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</span> <span class="intbl">
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<em>|a+h| − |a|</em>
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<strong>h</strong>
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</span></span></div>
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<div class="row"><span class="left">a=0:</span><span class="right"><span class="intbl"><span class="lim">lim</span>
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<strong>h→0</strong>
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</span> <span class="intbl">
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<em>|h| − |0|</em>
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<strong>h</strong>
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</span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="intbl"><span class="lim">lim</span>
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<strong>h→0</strong>
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</span> <span class="intbl">
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<em>|h|</em>
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<strong>h</strong>
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</span></span></div>
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</div>
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<p><b>In fact that limit does not exist!</b> To see why, let's compare left and right side limits:</p>
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<div class="tbl">
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<div class="row"><span class="left">From Left Side:</span><span class="right"><span class="intbl"><span class="lim">lim</span>
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<strong>h→0<sup><span class="hilite">−</span></sup></strong>
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</span> <span class="intbl">
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<em>|h|</em>
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<strong>h</strong>
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</span> = −1</span></div>
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<div class="row"><span class="left">From Right Side:</span><span class="right"><span class="intbl"><span class="lim">lim</span>
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<strong>h→0<sup><span class="hilite">+</span></sup></strong>
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</span> <span class="intbl">
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<em>|h|</em>
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<strong>h</strong>
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</span> = +1</span></div>
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</div>
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<p>The limits are different on either side, so the limit does not exist at x=0</p>
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<p> </p>
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<p class="larger center"> f(x) = |x| is not differentiable at x=0</p>
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</div>
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<p>A good way to picture this in your mind is to think:</p>
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<p class="center larger">As I zoom in, does the function tend to become a straight line?</p>
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<p class="center"><img src="images/differentiable.svg" alt="differentiable (zoomed is line) vs not differentiable (zoomed is pointy)" width="460" height="140" ></p>
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<p class="center">The absolute value function stays pointy at x=0 even when zoomed in.</p>
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<h2>Other Reasons</h2>
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<p>Here are a few more examples:</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="../sets/images/function-floor-graph.svg" alt="Floor function" width="172" height="170" ></td>
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<td> </td>
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<td>
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<p>The <a href="../sets/function-floor-ceiling.html">Floor and Ceiling Functions</a> are not differentiable at integer values, as there is a discontinuity at each jump. But they are differentiable elsewhere.</p></td>
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<td> </td>
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</tr>
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</tbody></table>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="images/x-1-3-slope.svg" alt="x^(1/3) slope" width="213" height="168" ></td>
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<td> </td>
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<td>
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<p>The Cube root function<b> x<sup>(1/3)</sup></b></p>
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<p>Its derivative is <b>(1/3)x<sup>-(2/3)</sup></b> (by the <a href="derivatives-rules.html">Power Rule</a>)</p>
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<p>At <b>x=0</b> the derivative is undefined, so x<sup>(1/3)</sup> is not differentiable, unless we exclude x=0.</p></td>
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</tr>
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</tbody></table>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>
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<p><img src="images/func-1-x.svg" alt="1/x graph" width="213" height="168" ></p></td>
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<td> </td>
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<td>
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<p>At <b>x=0</b> the function is not defined so it makes no sense to ask if they are differentiable there.</p>
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<p>To be differentiable at a certain point, the function must first of all be defined there!</p></td>
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</tr>
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</tbody></table>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><br>
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<img src="images/sin-1-x.svg" alt="sin (1/x) graph" width="213" height="168" ></td>
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<td> </td>
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<td>
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<p>As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is "heading towards".</p>
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<p>So it is not differentiable there.</p></td>
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</tr>
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</tbody></table>
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<p> </p>
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<h2>Different Domain</h2>
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<p>But we can change the domain!</p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/absolute-positive.svg" alt="absolute positive graph" width="101" height="102" ></p>
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<h3>Example: The function g(x) = |x| with Domain (0, +∞)</h3>
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<p>The domain is from <b>but not including</b> 0 onwards (all positive values).</p>
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<p><b>Which IS differentiable.</b></p>
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<p><i>And I am "absolutely positive" about that :)</i></p>
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<p class="larger">So the function <b>g(x) = |x| with Domain (0, +∞)</b> is differentiable.</p>
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<p>We could also restrict the domain in other ways to avoid x=0 (such as all negative Real Numbers, all non-zero Real Numbers, etc).</p>
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</div>
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<p> </p>
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<h2>Why Bother?</h2>
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<p>Because when a function is differentiable we can use all the power of calculus when working with it.</p>
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<h2>Continuous</h2>
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<p>When a function is differentiable it is also <a href="continuity.html">continuous</a>.</p>
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<p class="center larger">Differentiable <span class="largest">⇒</span> Continuous</p>
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<p>But a function can be <b>continuous but not differentiable</b>. For example the absolute value function is actually continuous (though not differentiable) at x=0.</p>
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<p> </p>
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<div class="questions">8925, 8926, 8930, 8931, 8927, 8928, 8929, 8932, 8933, 8934</div>
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<div class="related">
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<a href="derivatives-introduction.html">Introduction to Derivatives</a>
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<a href="index.html">Calculus Index</a>
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