Question

Differentiate the given function partially and choose the correct option:
$u = u(x,y) = \sin (y + ax) - {(y + ax)^2} \Rightarrow$
A. ${u_{xx}} = {a^2}.{u_{yy}}$.
B. ${u_{yy}} = {a^2}.{u_{xx}}$.
C. ${u_{xx}} = - {a^2}.{u_{yy}}$.
D. ${u_{yy}} = - {a^2}.{u_{xx}}$.

Answer 1

To solve this question, we will use the method of partial differentiation of a function with respect to any variable. Partial differentiation is the derivative of a dependent variable such that, z = f (x,y) with respect to independent variables x and y. We will differentiate the given function with respect to the variables given in the question and then we will get the relation between the derivatives.

Complete step-by-step answer:
Given that,
$u = u(x,y) = \sin (y + ax) - {(y + ax)^2}$. ………. (1)
Here u is a function with two independent variables x and y.
We will partially differentiate u with respect to x and y one by one to get the answer.
First, partially differentiating equation (1) with respect to x, we will get
$\Rightarrow {u_x} = \dfrac{d}{{dx}}u(x,y) \\\ \Rightarrow {u_x} = \dfrac{d}{{dx}}(\sin (y + ax)) - \dfrac{d}{{dx}}{(y - ax)^2} \\\ \Rightarrow {u_x} = a.\cos (y + ax) - 2(y - ax).( - a) \\\$
$\Rightarrow {u_x} = a\left( {\cos (y + ax) + 2(y - ax)} \right)$ ………… (2)
Again partially differentiating equation (2) with respect to x, we will get
$\Rightarrow {u_{xx}} = a\left( { - \sin (y + ax).a - 2a} \right)$
Now taking common – a from the above equation, we will get
$\Rightarrow {u_{xx}} = - {a^2}(\sin (y + ax) + 2)$. …….. (3)
Now, we will partially differentiate the equation (1) with respect to y,
$\Rightarrow {u_y} = \dfrac{{du(x,y)}}{{dy}}$
$\Rightarrow {u_y} = \dfrac{d}{{dy}}(\sin (y + ax)) - \dfrac{d}{{dy}}{(y - ax)^2} \\\ \Rightarrow {u_y} = \cos (y + ax) - 2(y - ax) \\\$
Again partially differentiating with respect to y, we will get
$\Rightarrow {u_{yy}} = - \sin (y + ax) - 2$
Now taking common – 1 from the above equation, we will get
$\Rightarrow {u_{yy}} = - (\sin (y + ax) + 2)$ ……….. (4)
Now, we will substitute the value of ${u_{yy}}$ in equation (3), we will get
$\Rightarrow {u_{xx}} = - {a^2}(\sin (y + ax) + 2) \\\ \Rightarrow {u_{xx}} = {a^2}{u_{yy}} \\\$
Here we can see that ${u_{xx}} = {a^2}{u_{yy}}$ for the given function u(x,y).
Therefore, the correct answer is option (A).

Note: Whenever we ask such a question, we have to remember some basic rules of partial differentiation. First, we will differentiate the given function with respect to the one of the given variables the times as required and then similarly we will differentiate with respect to another given variable. After that we will find out the relation between both of the derivatives obtained. Through this we will get the required answer.