Question

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

`"e"^(3x - 4)` at x = 2

Answer 1

Let, f(x) = `"e"^(3x - 4)`

∴ f(2) = `"e"^(3(2) - 4)` = e² and

f(2 + h) = `"e"^(3(2 + "h")-4` = e^3h+2  

By first principle, we get

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

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

= `lim_("h" -> 0) ("e"^(3"h" + 2) - "e"^2)/"h"`

= `lim_("h" -> 0) ("e"^(3"h") "e"^2 - "e"^2)/"h"`

= `lim_("h" -> 0) ("e"^2 ("e"^(3"h") - 1))/"h"`

= `"e"^2 lim_("h" -> 0) (("e"^(3"h") - 1)/(3"h")) xx 3`

= `3"e"^2 (1)     ...[because lim_(x -> 0) ("e"^("p"x) - 1)/("p"x) = 1]`

= 3e²