If f(x) = x , 2x , 4x , 8x , 16x , then f' ( 4 ) =

Step-by-step solution

Using the trigonometric identity x 2x 2^2x 2^ n-1 x = (2^n x) 2^n x For n=5 , the given function can be written as: f(x) = (32x) 32 x Differentiating with respect to x using the quotient rule: f'(x) = 32 x 32 (32x) - (32x) 32 x (32 x)^2 f'(x) = 32 (32x) x - (32x) x 32 ^2 x Substituting x = 4 : 32x = 32 4 = 8 Since (8 ) = 1 and (8 ) = 0 , we get: f' ( 4 ) = 32(1) ( 4 ) - 0 32 ^2 ( 4 ) f' ( 4 ) = 1 ( 4 ) f' ( 4 ) = cosec ( 4 ) Answer: cosec ( 4 )