If F(x) = f(x) g(x) and f ' (x) g ' (x) = c, then -
Text Solution
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(a,b,c)
F(x) = f(x). g(x)
Differentiate both sides w.r.t. x
F ' (x) = f ' (x) g(x) + g'(x) f(x) …(1)
F ' (x) = f ' (x) g'(x) 
F ' = c 
⇒
Again differentiate eq n (1)
F '' (x) = f '' (x) g(x) + g'(x) f ' (x) + g''(x) f(x) + f ' (x) g'(x)
F '' (x) = f '' (x) g(x) + g''(x) f(x) + 2g'(x) f ' (x) …(2)
Divide F(x) = f(x).g(x) on both sides
= 
+ 
+ 
= 
+ 
+ 
⇒
Again differentiate eq n (2)
F ''' (x) = f '' (x) g'(x) + f ''' (x) g(x) + g''(x) f ' (x) + g'''(x) f(x) + 0 {g'(x) f ' = c}
F ''' (x) = f ''' (x) g(x) + g'''(x) f(x) + f '' (x) g'(x) + g''(x) f ' (x)
f ' (x) g ' (x) = c differentiate both sides
f ' (x) g''(x) + g'(x) f '' (x) = 0 put it in above eq n
Dividing by F(x)
= 
+ 
⇒
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