Differentiate each of the following from first principle: `e^(3x)`

Explanation

Let `y = e^(3x)`.
Let `deltay` be an increment in y, corresponding to an increment `deltax` in x.
Then, `y + deltay = e^(3(x+deltx))`
`rArr deltay = e^(3(x+deltax)) - e^(3x)`
`rArr(deltay)/(deltax) = (e^(3(x+deltax)) - e^(3x))/(deltax)`
`rArr(dy)/(dx) = underset(deltax rarr 0)("lim") (e^(3(x+deltax))-e^(3x))/(deltax)`
`= underset(deltax rarr 0)("lim")3e^(3x).((e^(3deltax) - 1)/(3deltax))`
`= 3e^(3x). underset(xrarr0)("lim") ((e^(x) - 1)/(z))`, where `z = 3deltax`
`= (3e^(3x) xx 1)= 3e^(3x) , [ :' underset(x rarr 0)("lim") ((e^(z) - 1)/(z)) = 1]`
Hence, `d/(dx) (e^(3x)) = 3e^(3x)`.