How do you differentiate #f(x)= ( 4- cscx )/ (e^x + 2) # using the quotient rule?

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Answer 1 · 2016-05-06T03:12:55

#color(red)(f' (x)=((e^x*csc x*cot x+2*csc x*cot x -4*e^x+e^x*csc x))/((e^x+2)^2))#

Start from the given function

#f(x)=(4-csc x)/(e^x+2)#

The formula to be used in finding the derivative is

#d/dx(u/v)=(v d/dxu-ud/dxv)/v^2#

Let #u=(4-csc x)# and #v=(e^x+2)#

#f' (x)=d/dx((4-csc x)/(e^x+2))=((e^x+2) d/dx(4-csc x)-(4-csc x)d/dx(e^x+2))/((e^x+2)^2)#

#f' (x)=((e^x+2) (0-(-csc x*cot x))-(4-csc x)(e^x*1+0))/((e^x+2)^2)#

#color(red)(f' (x)=((e^x*csc x*cot x+2*csc x*cot x -4*e^x+e^x*csc x))/((e^x+2)^2))#

God bless....I hope the explanation is useful.