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If u =e^xyz then find the value of del^3u/delxdelydelz?

If u =e^xyz then find the value of del^3u/delxdelydelz?
 

Grade:12th pass

1 Answers

Vikas Dev Tara
13 Points
2 years ago
Given: u = {e^{xyz}}
We need to find \frac{{{\partial ^3}u}}{{\partial x\partial y\partial z}}.
Start with \dfrac{{\partial u}}{{\partial x}}.
\begin{array}{c}\dfrac{{\partial u}}{{\partial x}} = \dfrac{\partial }{{\partial x}}\left( {{e^{xyz}}} \right)\\ = yz{e^{xyz}}\end{array}
Now differentiate with respect to y.
\begin{aligned}\dfrac{{{\partial ^2}u}}{{\partial x\partial y}} &= \dfrac{\partial }{{\partial y}}\left( {yz{e^{xyz}}} \right)\\ &= z{e^{xyz}} + xy{z^2}{e^{xyz}}\end{aligned}
Now differentiate with respect to z.
$$\begin{aligned}\frac{{{\partial ^3}u}}{{\partial x\partial y\partial z}} &= \frac{\partial }{{\partial z}}\left( {z{e^{xyz}} + xy{z^2}{e^{xyz}}} \right)\\ &= {e^{xyz}} + xyz{e^{xyz}} + 2xyz{e^{xyz}} + {x^2}{y^2}{z^2}{e^{xyz}}\\ &= {e^{xyz}} + 3xyz{e^{xyz}} + {x^2}{y^2}{z^2}{e^{xyz}}\end{aligned}

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