Question

What is the derivative of `3^x`?

Answer 1

`3^xln3`

Explanation:

Begin by letting `y=3^x`

now take the ln of both sides.

`lny=ln3^xrArrlny=xln3`

differentiate `color(blue)"implicitly with respect to x"`

`rArr1/y dy/dx=ln3`

`rArrdy/dx=yln3`

now y = `3^xrArrdy/dx=3^xln3`

This result can be `color(blue)"generalised"` as follows.

`color(red)(bar(ul(|color(white)(a/a)color(black)(d/dx(a^x)=a^xlna)color(white)(a/a)|)))`

Answer 2

`d/dx(3^x)=ln(3)3^x`

Explanation:

`d/dx(3^x)`

`=d/dx(e^(x ln(3)))` (since `a^b=e^(b ln(a))`)

`=d/dx(x ln(3))d/(d(x ln(3)))(e^(x ln(3)))` (the chain rule)

`=ln(3)e^(x ln(3))`

`=ln(3)3^x`