document.write( "Question 515569: What is the difference quotient of f(x)=x^4 \n" ); document.write( "

Algebra.Com's Answer #344085 by drcole(72) $\"\"$ $\"About$

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The difference quotient of a function f(x) is:\r
\n" ); document.write( "\n" ); document.write( " $\"%28f%28x+%2B+h%29+-+f%28x%29%29%2F%28h%29\"$ \r
\n" ); document.write( "\n" ); document.write( "It measures the slope of the line that passes through the graph y = f(x) at x and x + h, where h is typically a very small number. If we are able to take the limit as h approaches 0, the resulting number is the derivative f'(x).\r
\n" ); document.write( "\n" ); document.write( "In this case, $\"f%28x%29+=+x%5E4\"$ , so\r
\n" ); document.write( "\n" ); document.write( " $\"f%28x+%2B+h%29+=+%28x+%2B+h%29%5E4\"$
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(multiplying out $\"%28x+%2B+h%29%5E4\"$ , with the intermediate steps skipped --- write out the product and distribute carefully)\r
\n" ); document.write( "\n" ); document.write( "Substituting into the formula for the difference quotient, we get:\r
\n" ); document.write( "\n" ); document.write( " $\"%28f%28x+%2B+h%29+-+f%28x%29%29%2F%28h%29\"$
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(substituting)
\n" ); document.write( " $\"%284%28x%5E3%29h+%2B+6%28x%5E2%29%28h%5E2%29+%2B+4x%28h%5E3%29+%2B+h%5E4%29%2F%28h%29\"$ (combining like terms)
\n" ); document.write( " $\"4+x%5E3+%2B+6%28x%5E2%29h+%2B+4x%28h%5E2%29+%2B+h%5E3+\"$ (canceling h)\r
\n" ); document.write( "\n" ); document.write( "Notice that as h goes to 0, all of the terms of the difference quotient go to 0 except for the first, leaving $\"4x%5E3\"$ as the derivative of $\"x%5E4\"$ . \n" ); document.write( "

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