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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
Solution
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`
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