What is (\partial)/(\partial x)((x^2+3y^3)e^{4x*y^2}) ?

The answer to (\partial)/(\partial x)((x^2+3y^3)e^{4x*y^2}) is 2xe^{4xy^2}+e^{4xy^2}*4y^2(x^2+3y^3)

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(\partial)/(\partial x)((x^2+3y^3)e^{4x*y^2})

\frac{\partial}{\partial x} ((x^{2} + 3 y^{3}) e^{4 x \cdot y^{2}})

2 x e^{4 x y^{2}} + e^{4 x y^{2}} \cdot 4 y^{2} (x^{2} + 3 y^{3})

Solution steps

\frac{\partial}{\partial x} ((x^{2} + 3 y^{3}) e^{4 x y^{2}})

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2} + 3 y^{3}, g = e^{4 x y^{2}}

= \frac{\partial}{\partial x} (x^{2} + 3 y^{3}) e^{4 x y^{2}} + \frac{\partial}{\partial x} (e^{4 x y^{2}}) (x^{2} + 3 y^{3})

\frac{\partial}{\partial x} (x^{2} + 3 y^{3}) = 2 x

\frac{\partial}{\partial x} (e^{4 x y^{2}}) = e^{4 x y^{2}} \cdot 4 y^{2}

= 2 x e^{4 x y^{2}} + e^{4 x y^{2}} \cdot 4 y^{2} (x^{2} + 3 y^{3})

\int \frac{x ^{3}}{49 - x ^{2}} d x

\int (x^{2} sin (y)) d y

\frac{d}{d x} (\frac{1}{sec ( x )})

\int \frac{1}{x 2 x} d x

\frac{d y}{d x} = \frac{8 x 4 - y ^{2}}{4 - y ^{2}}

What is (\partial)/(\partial x)((x^2+3y^3)e^{4x*y^2}) ?
The answer to (\partial)/(\partial x)((x^2+3y^3)e^{4x*y^2}) is 2xe^{4xy^2}+e^{4xy^2}*4y^2(x^2+3y^3)