document.write( "Question 1027271: [f(x)]^4 = (x + f(x))^3 and f(1)=2.
\n" ); document.write( "What is f'(1)? \n" ); document.write( "

Algebra.Com's Answer #642616 by robertb(5830) $\"\"$ $\"About$

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$\"%28f%28x%29%29%5E4+=+%28x+%2B+f%28x%29%29%5E3\"$ \r
\n" ); document.write( "\n" ); document.write( "We will use implicit differentiation (a direct application of the chain rule) to find f'(1).\r
\n" ); document.write( "\n" ); document.write( "==> 4 $\"%28f%28x%29%29%5E3\"$ *f'(x) = 3 $\"%28x%2Bf%28x%29%29%5E2\"$ *(1+f'(x)) by the chain rule.\r
\n" ); document.write( "\n" ); document.write( "==> 4 $\"%28f%281%29%29%5E3\"$ *f'(1) = 3 $\"%281%2Bf%281%29%29%5E2\"$ *(1+f'(1)) after letting x = 1.\r
\n" ); document.write( "\n" ); document.write( "<==> 4*2^3*f'(1) = 3(1+2)^2*(1+f'(1))
\n" ); document.write( "<==>32f'(1) = 27(1+f'(1))
\n" ); document.write( "==> 32f'(1) = 27 + 27f'(1)
\n" ); document.write( "==> 5f'(1) = 27, or\r
\n" ); document.write( "\n" ); document.write( "f'(1) = 27/5. \n" ); document.write( "

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