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Find the derivative of the function given by `f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)` and, hence, find `f^(prime)(1)dot`

Text Solution

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`f(x) = (1+x)(1+x^(2))(1+x^(4))(1+x^(8))`
`rArr "log"f(x) = "log"(1+x) + "log" (1+x^(2)) + "log"(1+x^(4)) + "log"(1+x^(8))`
Differentiate both side w.r.t.x
`(f'(x))/(f(x)) = (1)/(1+x) + (2x)/(1+x^(2)) + (4x^(3))/(1+x^(4)) + (8x^(7))/(1+x^(8))`
`rArr f'(x)= f(x)[(1)/(1+x) + (2x)/(1+x^(2)) + (4x^(3))/(1+x^(4)) + (8x^(7))/(1+x^(8))]`
`rArr f'(x)= (1+x) (1+x^(2)) (1+x^(4)) (1+x^(8)) [(1)/(1+x) + (2x)/(1+x^(2)) + (4x^(3))/(1+x^(4)) + (8x^(7))/(1+x^(8))]`
`therefore f'(1) = (2)(2)(2)(2)[(1)/(2) + (2)/(2) + (4)/(2) + (8)/(2)]`
`= 16 ((1)/(2) + 1 + 2 + 4) = 120`
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