Answer to Question #181810 in Calculus for desmond

Question

6. Without evaluating derivatives, which of the following functions have the same derivatives? f(x) = ln x, g(x) = ln 2x, h(x) = ln x 2 , k(x) = ln 10x 2

Answer 1

f(x) = ln x

g(x) = ln 2x

h(x) = ln x²

k(x) = ln 10x²

Without evaluating the derivatives we need to find that which functions have the same derivatives.

It means that we need to look at each of our functions and find out which functions have a constant as their difference. Only those functions will have same derivatives.

Now ln (xy) = ln x + ln y

and ln (xⁿ) = n ln (x)

By using the above properties the given functions can be written as

f(x) = ln x

g(x) = ln 2 +ln x

h(x) = 2 ln x

k(x) = ln 10 + 2 ln x

Now let us evaluate f(x) - g(x) and h(x) - k(x)

f(x) - g(x) = - ln 2

h(x) - k(x) = - ln 10

From the above equations we see that f(x) - g(x) and h(x) - k(x) have a constant as their difference.

Hence we conclude that f(x) , g(x) have same derivative and h(x) , k(x) have the same derivative.