How to find the derivative???

Question

asked Jul 20, 2022 103k views

1 Answer

Arth Du · Answer 1 · 2022-07-25T03:27:32+0000

Answer:

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

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
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

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