Differentiate y = t sin(t)/(1 + t)

Question

Differentiate y = t sin(t)/(1 + t)

 

 

ANSWER:  ((t² + t)(cos(t) + sin(t)) / (1 + t²) 

#13, p197

Answer 1

I believe your answer should have the denominator  (1+t)²  so could you check?

 

rewrite the original as:   

 

Y = T*SIN(T)*(1+T)^-1    

 

now take derivative as a product: f1*f2' + f1'*f2  where   f1 = T*SIN(T)   f2 = (1+T)^-1

 

Y' = T*SIN(T)*(-1(1+T)^-2)  +   d/dt( T*SIN(T)) * (1+T)^-1   (note: you have differentiate the prod T*SIN(T) ) 

 

       d/dt (T*SIN(T)) = (T*COS(T) + 1*SIN(T))

 

Y' =  - T*SIN(T)/(1+T)²  +  (T*COS(T) + 1*SIN(T)) / (1+T)    

 

now, get a common denom, multiply right side by (1+T)/(1+T)

 

Y' =  [ - T*SIN(T)  + (1+T)*(T*COS(T) + SIN(T)) ]  / (1+T)²

 

Y' = [ -T*SIN(T) + T*COS(T) + SIN(T)*1 + T²*COS(T) + T*SIN(T) ] / (1+T)²        (expanded numer. using FOIL)

 

( note: T*SIN(T) cancels, then factor out  COS(T) )

Y ' = [ COS(T)*(T² + T) + SIN(T) ] / (1+T)²        note: check your answer's denominator.