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

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

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(\partial)/(\partial x)(sin(xe^{x^2y}))

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

cos (x e^{x^{2} y}) (e^{y x^{2}} + 2 y e^{y x^{2}} x^{2})

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

= cos (x e^{x^{2} y}) (e^{y x^{2}} + 2 y e^{y x^{2}} x^{2})

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