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How to find the derivative???

How to find the derivative???-example-1
User Zerkey
by
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1 Answer

5 votes

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
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}

User Arth Du
by
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