Differential Calculus> If u =e^xyz then find the value of del^3u...

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

Ankita shree , 4 Years ago

Grade 12th pass

1 Answers

Vikas Dev Tara

Last Activity: 4 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}$