Question

Z=tan(y+ax) +(y-ax)^3/2∂^2z/∂x^2 =a^2 ∂^2z/∂y^2​

Answer 1

Answer:

Given z = tan (y + ax) – √(y-ax)

Differentiate partially w.r.t.x

zx = sec2(y + ax) a – [1/2√(y-ax)]×-a

= a sec2 (y + ax) + a/2√(y-ax)

Differentiate partially w.r.t.x

zxx = a×2 sec2 (y + ax) tan (y + ax)×a + (a/2)(-½)(y-ax)-3/2×-a

= 2a2 sec2(y+ax) tan (y+ax) + a2/4(y-ax)3/2

Differentiate partially w.r.t.y

zy = sec2(y+ax) – 1/2√(y-ax)

zyy = 2 sec2(y+ax) tan (y+ax) + 1/4(y-ax)3/2

zxx– a2zyy = 2a2 sec2(y+ax) tan (y+ax) + a2/4(y-ax)3/2 – a2[2 sec2(y+ax) tan (y+ax) + 1/4(y-ax)3/2]

= 0

Answer 2

Answer:

Function of two or more variables with respect to one of those variables is equal to the partial derivative of the same function with respect to the other variable.

Step-by-step explanation:

From the above question,

They have given :

The partial derivative of a function of two or more variables with respect to one of those variables is equal to the partial derivative of the same function with respect to the other variable.

So,

$∂^2z/∂x^2 = ∂^2z/∂y^2$

$a^2 tan(y + ax) + (3/2)a^2(y - ax)^2 + (3/2)(y - ax)^3 = a^2 sec^2(y + ax) + 3(y - ax)^2$

solving for a:

$a^2 sec^2(y + ax) - a^2 tan(y + ax) = - (3/2)a^2(y - ax)^2 + 3(y - ax)^2$

$a^2[sec^2(y + ax) - tan(y + ax)] = 3(y - ax)[(y - ax) - (1/2)a^2]$

$a^2[sec^2(y + ax) - tan(y + ax)] = 3(y - ax)[y - ax - (1/2)a^2]$

$a^2[sec^2(y + ax) - tan(y + ax)] = 3(y - ax)[y - ax - (1/2)a^2]$

$a^2[sec^2(y + ax) - tan(y + ax)] = 3(y - ax)[y - ax - (1/2)a^2]$

a = $√[(3(y - ax)[y - ax - (1/2)a^2])/(sec^2(y + ax) - tan(y + ax))]$

Therefore, the second partial derivative of z with respect to x is:

$∂^2z/∂x^2 = a^2 ∂^2z/∂y^2 = a^2 sec^2(y + ax) + 3(y - ax)^2$

where a  is given by

a = $√[(3(y - ax)[y - ax - (1/2)a^2])/(sec^2(y + ax) - tan(y + ax))]$

$∂^2z/∂x^2 = a^2 tan(y + ax) + (3/2)a^2(y - ax)^2 + (3/2)(y - ax)^3∂^2z/∂y^2 = a^2 sec^2(y + ax) + 3(y - ax)^2$

Partial derivatives are a type of derivative which is used to measure the rate of change of a function with respect to one of its variables, while keeping the other variables constant.

In other words, the partial derivative of a function of two or more variables with respect to one of those variables is equal to the partial derivative of the same function with respect to the other variable.

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