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

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

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(\partial)/(\partial x)(x^2y^2e^{4xy})

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

\frac{d}{d x} ((x \cdot sinh (x) + cos (x))^{5})

\frac{d}{d x} (\frac{x + 6}{x ^{3} + x - 8})

x^{2} - 2 x - 1

\frac{( s + 4 )}{s ^{2} + 4 s + 29}

\int 1

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