Q. 28 C4.7( 3 Votes )

Differentiate each of the following functions by the product by the product rule and the other method and verify that answer from both the methods is the same.

(3 sec x – 4 cosec x) ( – 2 sin x + 5 cos x)

Answer :

Let, y = (3 sec x – 4 cosec x)( – 2 sin x + 5 cos x)


y = – 6 sec x sin x + 15 sec x cos x + 8 sin x cosec x – 20cosec x cos x


y = – 6tan x + 15 + 8 – 20 cot x { tan x cos x = sin x}


y = – 6 tan x – 20 cot x + 23


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 = (3 sec x – 4 cosec x) and v = ( – 2 sin x + 5 cos 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 = (3 sec x – 4 cosec x)



Use the formula: &



…..equation 3


As, v = – 2 sin x + 5 cos 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 –




= 20(1 + cot2 x) – 6(1 + tan2 x)


[ 1 + tan2 x = sec2 x & 1 + cot2 x = cosec2 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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