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Question
Find the derivative of the following w. r. t. x by using method of first principle:
e2x+1
Solution
Let f(x) = e2x+1
∴ f(x + h) = `"e"^(2(x + "h") + 1)`
= `"e"^(2x + 2"h" + 1)`
By first principle, we get
f'(x) = `lim_("h" -> 0) ("f"(x + "h") - "f"(x))/"h"`
= `lim_("h" -> 0) ("e"^(2x + 2"h" + 1) - "e"^(2x +1))/"h"`
= `lim_("h" -> 0) "e"^(2x + 1) (("e"^(2"h") - 1))/"h"`
= `"e"^(2x + 1) (lim_("h" -> 0) ("e"^(2"h") - 1)/(2"h")) xx 2`
= 2 e2x+1 (1) ...`[lim_(x -> 0) ("e"^("p"x) - 1)/("p"x) = 1]`
= 2 e2x+1
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