Question

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

3^x 

Answer 1

Let f(x) = 3^x

∴ f(x + h) = 3^x+h

∴ f(x + h) – f(x) = 3^x+h – 3^x 

= 3^x . 3^h – 3^x 

= 3^x (3^h – 1)

By definition,

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

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

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

= 3^x. log 3   ...`[because  lim_(x -> 0) ("a"^x - 1)/x = log"a"]`