Suppose f(x)<\/em> =x3<\/sup> +x2<\/sup><\/em><\/p>\n and you would like to find the derivative.\u00a0 You might be thinking:<\/p>\n “Wait — I have a rule for each of those terms, separately.\u00a0 Can I just add them?\u00a0”<\/p>\n In other words, is f'(x)<\/em> = 3x2 <\/sup>+ 2x<\/i> ?\u00a0\u00a0 Yes it is.\u00a0 We could say this more officially:<\/p>\n If h(x) = f(x) + g(x) then h'(x) = f'(x) + g'(x).<\/strong><\/em><\/p>\n But all that means is that if you have functions added together, you can use their separate derivative formulas and then add them.\u00a0 And it works the same way for subtraction.<\/p>\n 2. Multiplying by\u00a0a Constant — Also as you would guess<\/strong><\/p>\n Given f(x) = 3sin(x)<\/em>…yes, it’s true: f'(x) = 3cos(x)<\/em>.\u00a0 Multiplying the original function by a constant just multiplies its derivative by the same constant.<\/p>\n There is an\u00a0intuitive way to see why this must be true:<\/p>\n Consider a function,\u00a0g(x) = a\u00b7f(x)<\/em>, where a>0.\u00a0\u00a0 How does the graph of g\u00a0compare to the graph of f?\u00a0[SAT fun fact!] \u00a0It is vertically stretched or vertically compressed, depending on whether a<\/em> is greater or less than one.\u00a0 Now, how would that affect the tangent line?\u00a0 A vertical stretch by a factor, a<\/em>, will increase the slope of that tangent line by that same factor, a<\/em>.<\/p>\n Here is an\u00a0illustration of what I mean:<\/p>\n On the left, you see\u00a0the graph of f(x) = \u221ax<\/em>\u00a0and its tangent line at x = 1.\u00a0 The slope of that tangent line is\u00a0.5.<\/p>\n And on the right, you see the graph of f(x) =\u00a03\u221ax<\/em> and its tangent line, also at x = 1.\u00a0 The graph has been stretched vertically by a factor of 3.\u00a0 And sure enough, the tangent line is 3 times as steep, or 1.5.<\/p>\n We’ll call that a “proof” and state the rule\u00a0officially:\u00a0 if g(x) = a\u00b7f(x) then g'(x) = a\u00b7f'(x)<\/em><\/strong>.<\/p>\n <\/p>\n 3. Multiplying and Dividing Functions: NOT what you would expect<\/strong><\/p>\n Well, it’s a bad new\/good news kind of thing…<\/p>\n Bad news:\u00a0you can’t just multiply the derivatives.\u00a0 In other words, if h(x) = f(x)\u00b7g(x), it\u00a0would be convenient\u00a0if \u00a0h'(x) = f'(x)\u00b7g'(x).\u00a0 But, alas.\u00a0 Though we don’t know what the rule is<\/span>, we know enough to see what it isn’t<\/span>.<\/p>\n All we have to do is let f(x)<\/em> = x3<\/sup><\/em> and let g(x) = x2<\/sup><\/em><\/em>.\u00a0 Then h(x) = f(x)\u00b7g(x) = x5\u00a0<\/sup><\/em><\/em>.<\/p>\n We also already know the derivatives: f'(x) =\u00a03x2<\/sup><\/em> and g'(x) = 2x<\/em>.\u00a0\u00a0When you multiply them, you get. 3x2<\/sup><\/em>\u00b72x = 6x3<\/sup><\/em>\u00a0 But we already know that h'(x) = 5x4<\/sup><\/em>.\u00a0 So the product of the derivatives did not match the derivative of the products.\u00a0 That’s the bad news. And a similar argument can be constructed to show that quotients don’t work that way either.<\/p>\n So what’s the good news?<\/p>\n 1.\u00a0 There are rules that handle these situations.\u00a0 They are called the product rule and the quotient rule.\u00a0 I am sure that your math teacher will be happy to teach them to you.<\/p>\n 2.\u00a0 This is where I pause and remind myself what these posts are for.\u00a0 I am trying to get you ready for your year of AP Physics.\u00a0 We may need the product rule and the quotient rule before the year is over, but we won’t need them right away.\u00a0 By the time we do need them, you will have seen them in math class.<\/p>\n On the other hand, there is one more way to combine functions that we will need almost immediately.\u00a0 So in the next post, we are going to look at something called “the chain rule”.<\/p>\n","protected":false},"excerpt":{"rendered":" In the last post, we compiled a small collection of derivative formulas, formulas which will enable you to find slopes of tangent lines.\u00a0 (Why you would want to do such a thing is a discussion that is coming soon.) You … Continue reading ![\"post17pic3\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post17pic3.jpg\")
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