document.write( "Question 1210577: Find the derivative of y=x using the first principle of differenciation
\n" ); document.write( "2) y=2x²—x \n" ); document.write( "

Algebra.Com's Answer #853737 by KMST(5334) $\"\"$ $\"About$

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I assume that in your class what was called the first principle of differentiation is that the derivative of a function $\"f%28x%29\"$ is
\n" ); document.write( "the limit of $\"%28f%28x%2BDELTA%28x%29%29-f%28x%29%29%2FDELTA%28x%29\"$ when $\"DELTA%28x%29\"$ tends to zero.
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\n" ); document.write( "For $\"y=x\"$ or $\"f%28x%29=x\"$ , $\"f%28x%2BDELTA%28x%29%29=x%2BDELTA%28x%29\"$ , $\"%28f%28x%2BDELTA%28x%29%29-f%28x%29%29%2FDELTA%28x%29\"$ $\"%22=%22\"$ $\"%28x%2BDELTA%28x%29-x%29%2FDELTA%28x%29\"$ $\"%22=%22\"$ $\"DELTA%28x%29%2FDELTA%28x%29=1\"$ ,
\n" ); document.write( "but $\"1\"$ does not depend on $\"DELTA%28x%29\"$ ,
\n" ); document.write( "so the limit of $\"1\"$ when $\"DELTA%28x%29\"$ tends to zero is $\"1\"$ and $\"highlight%28df%2Fdx=1%29\"$ or $\"highlight%28dy%2Fdx=1%29\"$
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\n" ); document.write( "For $\"y=2x%5E2%E2%80%94x\"$ or $\"f%28x%29=2x%5E2%E2%80%94x\"$ ,
\n" ); document.write( " $\"f%28x%2BDELTA%28x%29%29=2%28x%2BDELTA%28x%29%29%5E2-%28x%2BDELTA%28x%29%29%29\"$ $\"%22=%22\"$ $\"2%28x%5E2%2B2DELTA%28x%29%2Ax%2B%28DELTA%28x%29%29%5E2%29-x-DELTA%28x%29\"$ $\"%22=%22\"$ $\"2x%5E2%2B4DELTA%28x%29%2Ax%2B2%28DELTA%28x%29%29%5E2-x-DELTA%28x%29\"$ $\"%22=%22\"$ $\"2x%5E2%2B4DELTA%28x%29%2Ax%2B2%28DELTA%28x%29%29%5E2-x-DELTA%28x%29\"$ ,
\n" ); document.write( " $\"f%28x%2BDELTA%28x%29%29-f%28x%29%29\"$ $\"%22=%22\"$ $\"2x%5E2%2B4DELTA%28x%29%2Ax%2B2%28DELTA%28x%29%29%5E2-x-DELTA%28x%29-%282x%5E2-x%29\"$ $\"%22=%22\"$ $\"2x%5E2%2B4DELTA%28x%29%2Ax%2B2%28DELTA%28x%29%29%5E2-x-DELTA%28x%29-2x%5E2%2Bx%29\"$ $\"%22=%22\"$ $\"4DELTA%28x%29%2Ax%2B2%28DELTA%28x%29%29%5E2-DELTA%28x%29%29\"$ ,
\n" ); document.write( " $\"%28f%28x%2BDELTA%28x%29%29-f%28x%29%29%2FDELTA%28x%29\"$ $\"%22=%22\"$ $\"%284DELTA%28x%29%2Ax%2B2%28DELTA%28x%29%29%5E2-DELTA%28x%29%29%2FDELTA%28x%29\"$ $\"%22=%22\"$ $\"4x%2B2x%2B2DELTA%28x%29-1\"$ , and the limit of $\"4x%2B2DELTA%28x%29-1\"$ when $\"DELTA%28x%29\"$ tends to zero is $\"4x-1\"$ , so $\"highlight%28dy%2Fdx=4x-1%29\"$ \n" ); document.write( "

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