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F Z = Tan ( Y − a X ) + ( Y − a X ) 3 2 Then Show that ∂ 2 Z ∂ X 2 = a 2 ∂ 2 Z ∂ Y 2 - Applied Mathematics 1

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If `z=tan(y-ax)+(y-ax)^(3/2)` then show that `(del^2z)/(delx^2)= a^2 (del^2z)/(dely^2)`

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Solution

`z=tan(y-ax)+(y-ax)^(3/2)` 

Differentiate partially w.r.t.x, `(delz)/(delx)=sec^2 (y+ax).a+3/2(y-ax)^(1/2).(-a)`

∴ `(delz)/(delx)=alpha sec^2(y+ax)-(3a)/2(y-ax)^(1/2)` 

Again, differentiate partially w.r.t.x, 

`(del^2z)/(delx^2)=alpha^2[2sec^2(y+ax)tan(y+ax)-3/4(y-ax)^2]`

Differentiate (1) partially w.r.t.y, `(delz)/(dely)=sec^2(y+ax).1+3/2(y-ax)^(1/2)`

Again, differentiate partially w.r.t.y, `(del^2z)/(dely^2)=2sec^2(y+ax).tan(y)+ax-3/4(y-ax)^(-1/2)`

From (2)&(3), `(del^2z)/(delx^2)=a^2 (del^2z)/(dely^2)`

Concept: Partial Derivatives of First and Higher Order
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