What is (\partial ^2)/(\partial {u)(v)\partial v}(e^{{u}(v)v}) ?

The answer to (\partial ^2)/(\partial {u)(v)\partial v}(e^{{u}(v)v}) is

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(\partial ^2)/(\partial {u)(v)\partial v}(e^{{u}(v)v})

\frac{\partial ^{3}}{\partial u ( v ) \partial v ^{2}} (e^{u (v) v})

e^{v u (v)} v^{3} \frac{\partial}{\partial v} (u (v))^{2} + 2 e^{v u (v)} v^{2} u (v) \frac{\partial}{\partial v} (u (v)) + e^{v u (v)} v^{2} \frac{\partial ^{2}}{\partial v ^{2}} (u (v)) + e^{v u (v)} v u (v)^{2} + 4 e^{v u (v)} v \frac{\partial}{\partial v} (u (v)) + 2 e^{v u (v)} u (v)

Solution steps

\frac{\partial ^{3}}{\partial u ( v ) \partial v ^{2}} (e^{u (v) v})

\frac{\partial}{\partial u ( v )} (e^{u (v) v}) = e^{u (v) v} v

= \frac{\partial ^{2}}{\partial v ^{2}} (e^{u (v) v} v)

\frac{\partial}{\partial v} (e^{u (v) v} v) = e^{v u (v)} v (v \frac{\partial}{\partial v} (u (v)) + u (v)) + e^{v u (v)}

= \frac{\partial}{\partial v} (e^{v u (v)} v (v \frac{\partial}{\partial v} (u (v)) + u (v)) + e^{v u (v)})

\frac{\partial}{\partial v} (e^{v u (v)} v (v \frac{\partial}{\partial v} (u (v)) + u (v)) + e^{v u (v)}) = e^{v u (v)} v^{3} \frac{\partial}{\partial v} (u (v))^{2} + 2 e^{v u (v)} v^{2} u (v) \frac{\partial}{\partial v} (u (v)) + e^{v u (v)} v^{2} \frac{\partial ^{2}}{\partial v ^{2}} (u (v)) + e^{v u (v)} v u (v)^{2} + 4 e^{v u (v)} v \frac{\partial}{\partial v} (u (v)) + 2 e^{v u (v)} u (v)

= e^{v u (v)} v^{3} \frac{\partial}{\partial v} (u (v))^{2} + 2 e^{v u (v)} v^{2} u (v) \frac{\partial}{\partial v} (u (v)) + e^{v u (v)} v^{2} \frac{\partial ^{2}}{\partial v ^{2}} (u (v)) + e^{v u (v)} v u (v)^{2} + 4 e^{v u (v)} v \frac{\partial}{\partial v} (u (v)) + 2 e^{v u (v)} u (v)

\frac{d}{d x} ((a x + b) e^{2 x})

\int \frac{x + 4}{x ^{2} + 9} d x

\frac{\partial}{\partial y} (x y - 2 y)

f (x) g (x) h (x)

2 x^{2} + 8

What is (\partial ^2)/(\partial {u)(v)\partial v}(e^{{u}(v)v}) ?
The answer to (\partial ^2)/(\partial {u)(v)\partial v}(e^{{u}(v)v}) is