Question

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

log(2x + 1) at x = 2

Answer 1

let f(x) = log(2x + 1)

∴ f(2) = log(4 + 1) = log 5

f(2 + h) = log[2(2 + h) + 1] = log(5 + 2h)

By definition,

f'(2) = `lim_("h" -> 0) ("f"(2 + "h") - "f"(2))/"h"`

= `lim_("h" -> 0) (log(5 + 2"h") - log 5)/"h"`

= `lim_("h" -> 0) 1/"h" log ((5 + 2"h")/5)`

= `lim_("h" -> 0) (log(1 + (2"h")/5))/(((2"h")/5)) xx 2/5`

= `2/5 lim_("h" -> 0) (log(1 + (2"h")/5))/(((2"h")/5)`

= `2/5 xx 1    ...[because "h" -> 0, (2"h")/5 -> 0 and lim_(x -> 0) (log(1 + x))/x = 1]`

= `2/5`