# How do you implicitly differentiate # e^(3x)=sin(x+2y) #?

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To implicitly differentiate ( e^{3x} = \sin(x+2y) ), follow these steps:

- Differentiate both sides of the equation with respect to ( x ).
- Use the chain rule and product rule where necessary.
- Solve for ( \frac{dy}{dx} ) in terms of ( x ) and ( y ).

The implicit differentiation of the given equation ( e^{3x} = \sin(x+2y) ) yields:

[ 3e^{3x} = \cos(x+2y) \cdot (1 + 2y') ]

[ 3e^{3x} = \cos(x+2y) + 2y'\cos(x+2y) ]

[ 3e^{3x} = \cos(x+2y) + 2y'\cos(x+2y) ]

[ 3e^{3x} - \cos(x+2y) = 2y'\cos(x+2y) ]

[ y' = \frac{3e^{3x} - \cos(x+2y)}{2\cos(x+2y)} ]

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