Differentiate the following function by the product rule and the other method and verify that answer from both the methods is the same : `(3secx-4cosecx)(-2sinx+5cosx)`

Explanation

Product rule :


`u=3 \sec x-4 \cosec x ; v=-2 \sin x+5 \cos x`

`u'=3 \sec x \tan x+4cosec x cot x ; v'=-2 \cos x-5 \sin x`


Using the product rule :

`\frac{d}{d x}(u v)=u v'+v u'`

`\frac{d}{d x}[(3 sec x-4 cos e c x)(-2 sin x+5 cos x)]=(3 sec x-4 cosec x)(-2 cos x-5 sin x)+(-2 \sin x+5 cos x)(3 sec x tan x+4 cosec x cot x)`

`=-6+15 tan x+8 \cot x+20-6 \tan ^{2} x-8 \cot x-15 \tan x+20 \cot ^{2} x`

`=-6+20-6(\sec ^{2} x-1)+20(cosec^{2} x-1)`

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