Without evaluating derivatives, which of the functions 10 10 10 10 g(x) = 2x ,h(x) = x + 2, p(x) = x ? ln 2 have the same derivative as f(x) = x

Solution 25E Step 1 In this problem we have to find which of the given functions have same derivative as f(x) = x10 by using the fact that “Two differentiable functions, that differ by a constant always have the same derivative” … (1) First let us consider f(x)and g(x) Given f(x) = x , g(x) = 2x 10 10 10 f(x)g(x) = x 2x 10 = x f(x)g(x) = x where x is a function of xand is not a constant. Therefore f(x)and g(x)does not have same derivative. Step 2 Now consider f(x)and h(x) Given f(x) = x , h(x) = x +20 f(x)h(x) = x 10 (x +2) 10 10 = x x 2 = 2 f(x)h(x) = 2where 2is a constant. Therefore by ( 1) we get, f(x)and h(x)have same derivative.