Q. 27

Differentiate in two ways, using product rule and otherwise, the function

(1 + 2tan x)(5 + 4 cos x). Verify that the answers are the same.

Answer :

Let, y = (1 + 2 tan x)(5 + 4 cos x)


y = 5 + 4 cos x + 10 tan x + 8 tan x cos x


y = 5 + 4 cos x + 10 tan x + 8 sin x { tan x cos x = sin x}


Differentiating y w.r.t x –



Using algebra of derivatives, we have –



Use formula of derivative of above function to get the result.



…equation 1


Derivative using product rule –


We have to find dy/dx


As we can observe that y is a product of two functions say u and v where,


u = (1 + 2tan x) and v = (5 + 4cos x)


y = uv


As we know that to find the derivative of product of two function we apply product rule of differentiation.


By product rule, we have –


…equation 2


As, u = (1 + 2tan x)




…..equation 3 { }


As, v = 5 + 4cos x




…equation 4 { }


from equation 2, we can find dy/dx



using equation 3 & 4, we get –




sin x = tan x cos x , so we get –




[ 1 – sin2 x = cos2 x ]



Hence,


….equation 5


Clearly from equation 1 and 5 we observed that both equations gave identical results.


Hence, Results are verified


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