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hw 4 soln - Calculus for the Life Science I MAT1330A...

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Calculus for the Life Science I MAT1330A , MAT1330B, MAT1330E Assignment 4 Question 1 Compute the derivative of the following functions. a ) f ( x ) = - cos 2 (1 - 3 x ) b ) h ( t ) = 1 sin(3 t 2 ) c ) g ( x ) = x - e 2 x 1 - xe 2 x d ) f ( y ) = cos( radicalbig y 2 + 4) Don’t forget to simplify the results as much as possible. Solution: a ) f ( x ) = h 1 ( h 2 ( h 3 ( x ))), where h 3 ( x ) = 1 - 3 x , h 2 ( x ) = cos( x ) and h 1 ( x ) = - x 2 . We get f ( x ) = - 2 cos (1 - 3 x ) d d x parenleftbigg cos(1 - 3 x ) parenrightbigg = 2 cos (1 - 3 x ) sin(1 - 3 x ) d d x (1 - 3 x ) = - 6 cos(1 - 3 x ) sin(1 - 3 x ) . b ) We may use the rule for the derivative of a quotient to get h ( t ) = parenleftbigg d d t (1) parenrightbigg sin(3 t 2 ) - d d t sin(3 t 2 ) sin 2 (3 t 2 ) = - cos(3 t 2 ) d d t (3 t 2 ) sin 2 (3 t 2 ) = - 6 t cos(3 t 2 ) sin 2 (3 t 2 ) . We also have that h ( t ) = h 1 ( h 2 ( h 3 ( t ))), where h 3 = 3 t 2 , h 2 ( x ) = sin( x ) and h 1 ( x ) = x 1 . Thus, h ( t ) = - sin 2 (3 t 2 ) d d t sin(3 t 2 ) = - sin 2 (3 t 2 ) cos(3 t 2 ) d d t (3 t 2 ) = = - 6 t cos(3 t 2 ) sin 2 (3 t 2 ) . c ) We may use the rule for the derivative of a quotient to get g ; ( x ) = parenleftbigg d d x ( x - e 2 x ) parenrightbigg (1 - xe
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