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Question
Find the derivative of the following w. r. t. x by using method of first principle:
3x
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Solution
Let f(x) = 3x
∴ f(x + h) = 3x+h
∴ f(x + h) – f(x) = 3x+h – 3x
= 3x . 3h – 3x
= 3x (3h – 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"`
= 3x. log 3 ...`[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
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