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

Solution 24E Step 1 In this problem we have to find which of the given functions have same derivative 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) = ln x, g(x) = ln 2x f(x)g(x) = ln x ln 2x By using ln (mn) = ln m + ln nto ln 2xwe get, = ln x (ln 2+ln x) = ln x ln 2 ln x = ln 2 f(x)g(x) = ln 2where ln 2is a constant. Therefore by ( 1) we get, f(x)and g(x)have same derivative. Step 2 Now consider f(x)and h(x) 2 Given f(x) = ln x, h(x) = ln x 2 f(x)h(x) = ln x ln x a 2 By using ln x = a ln x to ln x we get, = ln x (2 ln x) = ln x 2 ln x = ln x f(x)h(x) = ln xwhere ln xis a function of xand is not a constant. Therefore f(x)and h(x)does not have same derivative. Step 3 Now consider f(x)and p(x) Given f(x) = ln x, p(x) = ln 10x 2 f(x)p(x) = ln x ln 10x 2 By using ln x = a ln x to ln 10x we get, = ln x (2 ln 10x) By using ln (mn) = ln m + ln nto ln 10xwe get, = ln x 2(ln 10+ ln x) = ln x2 ln 102 ln x f(x)p(x) = ln x2 ln 10where ln x2 ln 10is a function of xand is not a constant. Therefore f(x)and p(x)does not have same derivative.