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Calculus with Analytic Geometry by edwards & Penney soln ch7

Calculus with Analytic Geometry

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Section 7.1 C07S01.001: If f ( x ) = e 2 x , then f ( x ) = e 2 x · D x (2 x ) = 2 e 2 x . C07S01.002: If f ( x ) = e 3 x 1 , then f ( x ) = e 3 x 1 · D x (3 x 1) = 3 e 3 x 1 . C07S01.003: If f ( x ) = exp( x 2 ), then f ( x ) = exp( x 2 ) · D x ( x 2 ) = 2 x exp( x 2 ). C07S01.004: If f ( x ) = e 4 x 3 , then f ( x ) = e 4 x 3 · D x (4 x 3 ) = 3 x 2 e 4 x 3 . C07S01.005: If f ( x ) = e 1 /x 2 , then f ( x ) = e 1 /x 2 · D x (1 /x 2 ) = 2 x 3 e 1 /x 2 . C07S01.006: If f ( x ) = x 2 exp( x 3 ), then f ( x ) = 2 x exp( x 3 ) + x 2 · 3 x 2 exp( x 3 ) = (2 x + 3 x 4 )exp( x 3 ). C07S01.007: If g ( t ) = t exp( t 1 / 2 ), then g ( t ) = exp( t 1 / 2 ) + t · 1 2 t 1 / 2 exp( t 1 / 2 ) = 2 + t 2 exp( t 1 / 2 ). C07S01.008: If g ( t ) = ( e 2 t + e 3 t ) 7 , then g ( t ) = 7( e 2 t + e 3 t ) 6 (2 e 2 t + 3 e 3 t ). C07S01.009: If g ( t ) = ( t 2 1) e t , then g ( t ) = 2 te t ( t 2 1) e t = (1 + 2 t t 2 ) e t . C07S01.010: If g ( t ) = ( e t e t ) 1 / 2 , then g ( t ) = 1 2 ( e t e t ) 1 / 2 ( e t + e t ). C07S01.011: If g ( t ) = e cos t = exp(cos t ), then g ( t ) = ( sin t )exp(cos t ). C07S01.012: If f ( x ) = xe sin x = x exp(sin x ), then f ( x ) = exp(sin x ) + ( x cos x )exp(sin x ) = e sin x (1 + x cos x ) . C07S01.013: If g ( t ) = 1 e t t , then g ( t ) = te t (1 e t ) t 2 = te t + e t 1 t 2 . C07S01.014: If f ( x ) = e 1 /x , then f ( x ) = 1 x 2 e 1 /x . C07S01.015: If f ( x ) = 1 x e x , then f ( x ) = ( 1) e x (1 x ) e x ( e x ) 2 = 1 1 + x e x = x 2 e x . C07S01.016: If f ( x ) = exp( x ) + exp( x ), then f ( x ) = 1 2 x 1 / 2 exp ( x ) 1 2 x 1 / 2 exp ( x ) = exp( x ) exp( x ) 2 x . C07S01.017: If f ( x ) = exp( e x ), then f ( x ) = e x exp( e x ). C07S01.018: If f ( x ) = ( e 2 x + e 2 x ) 1 / 2 , then f ( x ) = 1 2 ( e 2 x + e 2 x ) 1 / 2 ( 2 e 2 x 2 e 2 x ) = e 2 x e 2 x e 2 x + e 2 x . 1
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C07S01.019: If f ( x ) = sin(2 e x ), then f ( x ) = 2 e x cos(2 e x ). C07S01.020: If f ( x ) = cos( e x + e x ), then f ( x ) = ( e x e x )sin( e x + e x ). C07S01.021: If f ( x ) = ln(3 x 1), then f ( x ) = 1 3 x 1 · D x (3 x 1) = 3 3 x 1 . C07S01.022: If f ( x ) = ln(4 x 2 ), then f ( x ) = 2 x x 2 4 . C07S01.023: If f ( x ) = ln (1 + 2 x ) 1 / 2 , then f ( x ) = 1 2 · 2(1 + 2 x ) 1 / 2 (1 + 2 x ) 1 / 2 = 1 1 + 2 x . C07S01.024: If f ( x ) = ln (1 + x ) 2 , then f ( x ) = 2(1 + x ) (1 + x ) 2 = 2 1 + x . C07S01.025: If f ( x ) = ln ( x 3 x ) 1 / 3 = 1 3 ln( x 3 x ), then f ( x ) = 3 x 2 1 3( x 3 x ) . C07S01.026: If f ( x ) = ln (sin x ) 2 = 2ln(sin x ), then f ( x ) = 2cos x sin x = 2cot x . C07S01.027: If f ( x ) = cos(ln x ), then f ( x ) = sin(ln x ) x . C07S01.028: If f ( x ) = (ln x ) 3 , then f ( x ) = 3(ln x ) 2 x . C07S01.029: If f ( x ) = 1 ln x , then (by the reciprocal rule) f ( x ) = 1 x (ln x ) 2 . C07S01.030: If f ( x ) = ln(ln x ), then f ( x ) = 1 x ln x . C07S01.031: If f ( x ) = ln x ( x 2 + 1) 1 / 2 , then f ( x ) = ( x 2 + 1) 1 / 2 + x 2 ( x 2 + 1) 1 / 2 x ( x 2 + 1) 1 / 2 = 2 x 2 + 1 x ( x 2 + 1) . C07S01.032: If g ( t ) = t 3 / 2 ln( t + 1), then g ( t ) = 3 2 t 1 / 2 ln( t + 1) + t 3 / 2 t + 1 = t 1 / 2 [2 t + 3ln( t + 1) + 3 t ln( t + 1)] 2( t + 1) . C07S01.033: If f ( x ) = lncos x , then f ( x ) = sin x cos x = tan x . C07S01.034: If f ( x ) = ln(2sin x ) = (ln2) + ln(sin x ), then f ( x ) = cos x sin x = cot x . C07S01.035: If f ( t ) = t 2 ln(cos t ), then f ( t ) = 2 t ln(cos t ) t 2 sin t cos t = t [2ln(cos t ) t tan t ]. C07S01.036: If f ( x ) = sin(ln2 x ), then f ( x ) = [cos(ln2 x )] · 2 2 x = cos(ln2 x ) x . 2
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C07S01.037: If g ( t ) = t (ln t ) 2 , then g ( t ) = (ln t ) 2 + t · 2ln t t = (2 + ln t )ln t. C07S01.038: If g ( t ) = t 1 / 2 [cos(ln t )] 2 , then g ( t ) = 1 2 t 1 / 2 [cos(ln t )] 2 + 2 t 1 / 2 [cos(ln t )] · sin(ln t ) t = [cos(ln t )][cos(ln t ) 4sin(ln t )] 2 t 1 / 2 .
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