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rules.html
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<!DOCTYPE html>
<html lang="en-us">
<head>
<title>☆【 Differentiation Website 】☆</title>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/bootstrap@5.1.3/dist/css/bootstrap.min.css" integrity="sha384-1BmE4kWBq78iYhFldvKuhfTAU6auU8tT94WrHftjDbrCEXSU1oBoqyl2QvZ6jIW3" crossorigin="anonymous">
<link rel="stylesheet" type="text/css" href="stylesheet.css">
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<body>
<div class="container">
<div class="row"><div id="websiteheader">
<div class="col-lg-12"><div id="toptitle">
<h3><strong>></strong> DIFFERENTIATION</h1>
<h6>Created by Joanne Lee<span id="blink">|</span></h6>
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<h4><strong>DIRECTORY</strong></h4>
<hr>
<ul>
<li><button><a href="index.html" id = "a"><strong>></strong> What is differentiation?</a></button></li>
<li><button><a href="rules.html" id = "b"><strong>></strong> Differentiation rules</a></button></li>
<li><button><a href="game.html" id = "c"><strong>></strong> Game (practice)</a></button></li>
</ul>
</div></div>
<div class="row"><div id="box2">
<h3><strong>Differentiation Rules:</strong></h3>
<span id="colorsquare1">.</span><span id="colorsquare2">.</span><span id="colorsquare3">.</span>
<br>
<p><div id = box3><h5 id = "white"><i>Basic Differentiation Rules</i><hr></h5><br>
<ul>
<li><b id = "red">1. Derivative of a constant function</b><br><b>></b> The derivative of a constant function is 0.<br><b>></b> ex. if f(t) = -34, f'(t) = 0</li><br>
<li><b id = "yellow">2. The Single Variable Rule</b><br><b>></b> The derivative of x is 1.<br><b>></b> ex. if f(x) = x, f'(x) = 1<br><b>></b> ex. if s(t) = t, s'(t) = 1</li><br></li>
<li><b id = "blue">3. The Power Rule</b><br><b>></b> If n is a rational number, then the function x<sup>n</sup> is differentiable and (d/dx)[x<sup>n</sup>] = n * x<sup>n-1</sup>.<br><b>></b> ex. if f(t) = x<sup>4</sup>, f'(t) = 4x<sup>3</sup></li><br>
<li><b id = "yellow">4. The General Power Rule</b><br><b>></b> The derivative of the term ax<sup>n</sup>, where a and n are real numbers, is <br>(a * n)x<sup>n-1</sup>.<br><b>></b> ex. if g(x) = -5t<sup>3</sup>, g'(x) = -15t<sup>2</sup></li><br>
<li><b id = "red">5. The Sum & Difference Rules</b><br><b>></b> The derivative of a sum or difference is the sum or difference of the derivatives.<br><b>></b> (d/dx)[f(x) + g(x)] = f'(x) + g'(x)<br><b>></b> (d/dx)[f(x) - g(x)] = f'(x) - g'(x)</li>
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</ul>
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</p>
<p><div id = box6><h5 id = "white"><i>The Product and Quotient Rules</i></h5><br>
<ul>
<li><b>The Product Rule:</b> If a function is the <u>product</u> of two differential functions (ex. f(x) and g(x)), the derivative is<br><br>f'(x) * g(x) + f(x) * g'(x)</li><br>
<li><b>The Quotient Rule:</b> If a function is the <u>quotient</u> of two differential functions (ex. f(x) and g(x)), the derivative is<br><br><u>f'(x) * g(x) - f(x) * g'(x)</u><br>[g(x)]<sup>2</sup></li>
</ul>
</div>
</p>
<p><div id = box7><h5 id = "white"><i>The Chain Rule</i></h5><br>
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<div class="col-lg-5" id = "right-align-2">
If f and g are both differentiable, then f(g(x)) is also differentiable and <br>f'(g(x)) * g'(x).
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<div class="col-lg-7">
<b>Example:</b> Find f'(x) if f(x) = (5x+3)<sup>2</sup>
<ul id = "right-align">
<li>1. f'(x) = 2(5x+3) * (5x+3)'</li>
<li>2. f'(x) = 2(5x+3)(5)</li>
<li>3. f'(x) = 10(5x+3)</li>
</ul>
</div>
</div>
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</p>
<p><div id = box5><h5 id = "black"><i>Implicit Differentiation</i></h5><br>
<b>Basic facts to consider:</b>
In implicit differentiation, you will have a (dy/dx) for each y in the original function or equation. Isolate the (dy/dx). If you are taking the second derivative, you will often substitute the expression you found for the first derivative somewhere in the process.<br>
<ul>
<li><b>></b> Differentiation is taking place with respect to x.</li>
<li><b>></b> When differentiating terms involving x alone, differentiate as usual.</li>
<li><b>></b> When differentiating terms involving y, apply the chain rule (it is assumed that y is defined implicitly as differentiable function of x.</li>
</ul>
<div id = box6>
<b>How to solve an implicit differentiation problem:<br></b>
<ul>
<li>1. Differentiate both sides with respect to x. Multiply by dy/dx every time you differentiate an expression containing y (apply chain rule).</li>
<li>2. Isolate dy/dx by performing the necessary steps to transfer all of the non-dy/dx terms onto one side of the equation with the dy/dx on the other.</li>
<li>3. Factor out dy/dx if necessary.</li>
<li>4. Solve for dy/dx and find the answer.</li>
</ul>
</div>
</div>
</p>
<p><div id = box9><h5 id = "black"><i>Derivatives of Inverse Functions</i></h5><br>
A function g is the <u>inverse</u> function of function f if f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of f. The function g(x) is denoted by f<sup>-1</sup>(x).
<br><br><u>To find the derivative of an inverse function:</u> Let f and g be inverse functions, such that f(g(x)) = x = g(f(x)) where f(a) = b and <br>g(b) = a.
<br><br>Finding g'(b) for a point (a, b) on f'(x):
<ul id = "right-align">
<li>1. Find the value of f'(x)</li>
<li>2. If you are only given b, set b = f(x) to find a.</li>
<li>3. Find f'(a)</li>
<li>4. g'(b) = 1/(f'(a))</li><br>
<li><b>></b> Essentially: (f<sup>-1</sup>)'(x) = <u> 1 </u><br> f'(f<sub>-1</sub>(x))</li>
</ul>
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</p>
<p><div id = "box8"><h5 id = "white"><i id = "glitch">Derivatives List</i></h5><br>
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<div class="col-lg-6" id="right-align-2">
<b id="right-align-2">Basic Derivatives:</b>
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<ul id="right-align-2">
<li><b>></b> (sin<i>x</i>)' = cos<i>x</i></li>
<li><b>></b> (cos<i>x</i>)' = -sin<i>x</i></li>
<li><b>></b> (tan<i>x</i>)' = sec<sup>2</sup><i>x</i></li>
<li><b>></b> (cot<i>x</i>)' = -csc<sup>2</sup><i>x</i></li>
<li><b>></b> (sec<i>x</i>)' = sec<i>x</i>tan<i>x</i></li>
<li><b>></b> (csc<i>x</i>)' = -csc<i>x</i>cot<i>x</i></li>
<li><b>></b> (ln<i>u</i>)' = 1/<i>u</i>(du/dx)</li>
<li><b>></b> (e<sup><i>u</i></sup>)' = e<sup><i>u</i></sup>(du/dx)</li><br>
<li id = "red">* Where <i>u</i> is a function of x, and <i>a</i> is a constant</li>
</ul>
</div>
</div>
<div class="col-lg-6" id="right-align">
<b> Derivatives of Trig Functions:</b>
<div id = "yellow">
<ul id="right-align">
<li><b>></b> (sin<sup>-1</sup><i>x</i>)' = <u> 1 </u><br> <b>√</b>(1-x<sup>2</sup>)</li>
<li><b>></b> (cos<sup>-1</sup><i>x</i>)' = <u> </u> <u> 1 </u><br> <b>√</b>(1-x<sup>2</sup>)</li>
<li><b>></b> (tan<sup>-1</sup><i>x</i>)' = <u> 1 </u><br> (1+x<sup>2</sup>)</li>
<li><b>></b> (cot<sup>-1</sup><i>x</i>)' = <u> </u> <u> 1 </u><br> (1+x<sup>2</sup>)</li>
<li><b>></b> (sec<sup>-1</sup><i>x</i>)' = <u> </u> <u> 1 </u><br> |x|*<b>√</b>(x<sup>2</sup>-1)</li>
<li><b>></b> (csc<sup>-1</sup><i>x</i>)' = <u> </u> <u> </u> <u> 1 </u><br> |x|*<b>√</b>(x<sup>2</sup>-1)</li>
</ul>
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