If h(x)=f(x)g(x), what is h’(x)?

 

Our goal is to test different examples, looking for solutions that we think might work until we get down to only one possible candidate solution that fits all tests, and then test it two more times.

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What actually happened--it wasn't a matter of thinking something would work.  It was trying something to see what would happen, and proving what was wrong with it.

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If the equation we write for the Product Rule is correct, it will work for any pairing of f(x) and g(x).  We could do something like test f(x)=x^2 and g(x)=y^3, and we would know the answer because h(x)=y^5 and we know the derivative of a monomial, so h'(x)=5y^4.

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We have yet another option--we can choose for one of the two functions to be "1".  This is "tricky", sometimes it forces the correct answer to step forward fairly quickly, and sometimes it fails miserable.

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Let f(x)=2x and g(x)=1.

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Our first idea from Black Magic to use is the idea that the derivative, h'(x), should have pieces or things coming from the original function, h(x).  Does it make sense that the derivative of the product of two functions might have the derivatives of those two 

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WORK HERE STOPPED -- NOT HAVING LATEX MAKES THIS UNTENABLE.

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