मान लीजिए कि `f(x +y) = f(x) f(y)` और `f(x) = 1 + xg(x) phi (x)` है, जहाँ `underset(xrarr0)"lim"g(x) = a`और `underset(xrarr0)"lim" phi (x) = b` है । तब `f'(x)` किसके बराबर है ?

Options

A. `1+ab(x)`
B. `1+ab`
C. `ab`
D. `abf(x)`

Explanation

माना `f(x+y) = f(x) f(y)`
और `f(x) = 1 + x g(x). phi(x)`
जहाँ `underset(xrarr0)("lim") g(x)= a`
और `underset(xrarr0)("lim") phi(x) = b`
हम जानते है ,
`f'(x) = underset(yrarr0)("lim") (f(x+y) - f(x))/(y)`
` = underset(yrarr0)("lim")(f(x).f(y) -f(x))/(y)`
` = underset(yrarr0)("lim") (f(x)[f(y)-1])/(y)`
` = underset(y rarr0)("lim")f(x) [(1+yg.(y)phi (y) - 1)/(y)]`
` = f(x) underset(yrarr0)("lim") g(y) underset(yrarr0)("lim") phi(y)`
` = f(x) .a.b= ab f(x)`