Question

Find the derivative of the following w. r. t. x by using method of first principle:

e^2x+1

Answer 1

Let f(x) = e^2x+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 e^2x+1 (1)  ...`[lim_(x -> 0) ("e"^("p"x) - 1)/("p"x) = 1]` 

= 2 e^2x+1