Question

What's the derivative of `arctan(6^x)`?

Answer 1

`y' = ( ln(6) 6^x)/( 1 + 6^{2x} )`

Explanation:

`y=arctan(6^x)`
`tan y=6^x`
`sec^2 y \ y' = (6^x)'`

`z = 6^x`
`ln z = x ln(6)`
`1/z z' = ln(6)`
`z' = ln(6) 6^x`

`\implies sec^2 y \ y' = ln(6) 6^x`

`y' = ( ln(6) 6^x)/( sec^2 y)`

using `tan^2 +1 = sec^2`

`y' = ( ln(6) 6^x)/( 1 + 6^{2x} )`