Find the indicated partial derivative. $u = x ^ { e } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a 1 , b 2 , c 3$
A) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = c b ( b - 1 ) c ( a - 1 ) ( a - 2 ) x ^ { c - 1 } y ^ { b - 2 } z ^ { a - 3 }$
B) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$
C) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a b ( b - 1 ) c ( c - 1 ) ( c - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$
D) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a c b ( a - 1 ) ( a - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$
E) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { b - 1 } y ^ { c - 2 } z ^ { a - 3 }$

Question

Find the indicated partial derivative. $u = x ^ { e } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$
A) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = c b ( b - 1 ) c ( a - 1 ) ( a - 2 ) x ^ { c - 1 } y ^ { b - 2 } z ^ { a - 3 }$
B) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$
C) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a b ( b - 1 ) c ( c - 1 ) ( c - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$
D) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a c b ( a - 1 ) ( a - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$
E) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { b - 1 } y ^ { c - 2 } z ^ { a - 3 }$

Quizplus · Accepted Answer

We need to use the following formula for the partial derivative of a function with three variables:
$$\frac{\partial^6 u}{\partial x\partial y^2 \partial z^3}= \frac{\partial^6}{\partial x\partial y^2 \partial z^3} (x^ey^bz^c)$$
Taking the partial derivative of the function 6 times with respect to x, y, and z, we get:
$$\frac{\partial^6 u}{\partial x\partial y^2 \partial z^3}= e(e-1)(a-2)(a-3)(b-2)(b-3)c(c-1)(c-2)x^{e-a}y^{b-4}z^{c-6}$$
Substituting the values of a, b, and c, we get:
$$\frac{\partial^6 u}{\partial x\partial y^2 \partial z^3}= abc(b-1)(b-2)(c-1)(c-2)(c-3)x^{a-1}y^{b-2}z^{c-3}$$
Therefore, the correct choice is C.

Find the Indicated Partial Derivative u=xeybzc;∂6u∂x∂y2∂z3,a>1,b>2,c>3u = x ^ { e } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3u=xeybzc;∂x∂y2∂z3∂6u​,a>1,b>2,c>3

Find the Indicated Partial Derivative $u = x ^ { e } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$