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posted by  Cal Help on 11/4/2009 5:17:24 PM  |  status: Live  |  Earned Karma: 0

Differentiation Rules

Course Textbook Chapter Problem Needs by
Calculus Essential Calculus Early Transcendentals (1st) by Stewart 3 N/A 11/5/2009 at 6:00:00 PM
Question Details:
δ2. Given the function ln y=eysinx
a) (3 marks) Find the first derivative.
b) (2 marks) Find an equation of the tangent line at the point (0, 1).
c) (1 mark) Illustrate part b) by plotting the tangent lines and the
function on the same set of axes.

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posted by anonymousxyz on 11/4/2009 6:10:06 PM  |  status: Live
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Response Details:


ln y  =  ey sin x  =>

d / dx ( ln y )  =  d / dx ( ey sin x )  =>

( 1 / y )( dy / dx )  =  ( ey sin x )( dy / dx )  +  ey cos x  =>

( ey sin x )( dy / dx )  -  ( 1 / y )( dy / dx )  =  - ey cos x  =>

( ey sin x  -  1 / y )( dy / dx )  =  - ey cos x  =>

dy / dx  =  ( - ey cos x ) / ( ey sin x  -  1 / y )  =>

dy / dx  =  ( - ey cos x ) / [ ( yey sin x  -  1 ) / y ]  =>

dy / dx  =  ( - yey cos x ) / ( yey sin x  -  1 )


The slope of the line tangent to the curve at the point ( 0, 1 ) is:


m  =  y'( 0, 1 )  =  ( -e ) / ( -1 )  =  e


The equation of the tangent line is:


y  =  mx  +  b  =  ex  +  b


Find b:


y( 0 )  =  1  =>

b  =  1  =>

y  =  ex  +  1
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