If u = ( x^2 + y^2 + z^2 )^ 3/2 , then ( u x )^2 + ( u y )^2 + ( … - Vidyadip

MCQ

If $u = {({x^2} + {y^2} + {z^2})^{3/2}}$, then ${\left( {{{\partial u} \over {\partial x}}} \right)^2} + {\left( {{{\partial u} \over {\partial y}}} \right)^2} + {\left( {{{\partial u} \over {\partial z}}} \right)^2} = $

A
$9u$
✓
$9{u^{4/3}}$
C
$9{u^2}$
D
${u^{4/3}}$

✓

Answer

Correct option: B.

$9{u^{4/3}}$

b
(b) $\frac{{\partial u}}{{\partial x}} = \frac{3}{2}{({x^2} + {y^2} + {z^2})^{1/2}}.2x$

$\therefore $ ${\left( {\frac{{\partial u}}{{\partial x}}} \right)^2} = \frac{9}{4}({x^2} + {y^2} + {z^2})4{x^2} = 9{x^2}({x^2} + {y^2} + {z^2})$

$\therefore $ ${\left( {\frac{{\partial u}}{{\partial x}}} \right)^2} + {\left( {\frac{{\partial u}}{{\partial y}}} \right)^2} + {\left( {\frac{{\partial u}}{{\partial z}}} \right)^2}$

$= 9\,({x^2} + {y^2} + {z^2})\,({x^2} + {y^2} + {z^2})$

$= 9\,{({x^2} + {y^2} + {z^2})^2}$ = $9.{u^{4/3}}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $A=\left[\begin{array}{ccc}1 & -2 & 4 \\ 2 & -1 & 3 \\ 4 & 2 & 0\end{array}\right]$ is the adjoint of a square matrix $B$, then $B^{-1}$ is equal to

The mean and variance of a binomial distribution are $6$ and $4$. The parameter $n$ is

Which of the following is correct?

Determinant is a square matrix
Determinant is a number associated to a matrix
Determinant is a number associated to a square matrix
None of these

Let $A^{-1}=\left[\begin{array}{lll}1 & 2017 & 2 \\ 1 & 2017 & 4 \\ 1 & 2018 & 8\end{array}\right]$ Then, $|2 A|-\left|2 A^{-1}\right|$ is equal to

Let $f(x)=\sqrt{\lim _{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}$ be differentiable in $(-\infty, 0) \cup(0, \infty)$ and $f(1)=1$. Then the value of $ea$, such that $f(a)=0$, is equal to.............

Mark the correct alternative in the following question:
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If $\frac{\text{P(X = r})}{\text{P(X = n} -\text{r})}$ is independent of n and r, then p equals:

$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{5}$
$\frac{1}{7}$

The velocity of a particle at time $t$ is given by the relation $v = 6t - {{{t^2}} \over 6}$. The distance traveled in $3$ seconds is, if $s = 0$ at $t = 0$

The number of real solutions of $x^{7}+5 x^{3}+3 x+1=$ $0$ is equal to............

The cost function at American Gadget is $C(x) = x^3 - 6x^2 + 15x$ $(x$ in thousands of units and $x > 0)$ The production level at which average cost is minimum is

The integral $\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x$ equals (for some arbitrary constant $K$ )