Question

How to find the derivative???

Answer 1

`y'= (2)/(3)e^{2x + e^(2x)/3}`

Explanation:

We have
`e^(2x) = \ln(y^3)` and we want the implicit differentiation
`(dy)/(dx)f(x) = y'`

Therefore, we want to separate the
`y` from the equation.

`\ln(y^3) =e^(2x) \implies 3 \ln(y) = e^(2x) \implies \ln(y) = (e^(2x) )/(3)`

`\implies e^(\ln(y)) = e^{e^(2x)/3} \implies \boxed{y = e^{e^(2x)/3}}`

Now we can calculate the derivative. By the chain rule,

`y' = (d)/(dx)\left(e^{e^(2x)/3}\right) = e^{e^(2x)/3}(d)/(dx)\left((e^(2x))/(3)\right)`

Now

`(d)/(dx)\left((e^(2x))/(3)\right) =(1)/(3)(d)/(dx)\left(e^(2x)\right) = (2)/(3)e^(2x)`

Therefore

`y' = e^{e^(2x)/3}\cdot (2)/(3)e^(2x) = (2)/(3)e^{2x + e^(2x)/3}`