Question

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

`2^(3x + 1)` at x = 2

Answer 1

Let f(x) = `2^(3x + 1)` 

∴ f(2) = `2^(3(2) + 1)` = 2⁷ and

f(2 + h) = `2^(3(2 + "h") + 1) = 2^(3"h" + 7)`

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) (2^(3"h" + 7) - 2^7)/"h"`

= `lim_("h" -> 0) (2^(3"h") * 2^7 - 2^7)/"h"`

= `lim_("h" -> 0) (2^7 (2^(3"h") - 1))/"h"`

= `2^7 lim_("h" -> 0) ((2^(3"h") - 1)/(3"h")) xx 3`

= `2^7 (log 2) xx 3   ...[lim_(x -> 0) (("a"^("p"x) - 1)/("p"x)) = log "a"]`

= 384 log 2