Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

# Verify uyuy∂2u∂x∂y=∂2u∂y∂x for u = x3 + 3x2 y2 + y3. - Business Mathematics and Statistics

Sum

Verify (del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x) for u = x3 + 3x2 y2 + y3.

#### Solution

Given u = x3 + 3x2 y2 + y3

Differentiating partially with respect to 'y' we get,

(del "u")/(del "y") = 0 + 3x^2 (2y) + 3y^2 = 6x^2y + 3y^2

Differentiating again partially with respect to 'x' we get,

(del^2 "u")/(del x del "y") = 6y(2x) + 0 = 12xy   ...(1)

Differentiating 'u' partially with respect to 'x' we get,

(del "u")/(del "x") = 3x2 + 3y2 (2x) + 0 = 3x2 + 6xy2

Differentiating again partially with respect to 'y' we get,

(del^2 "u")/(del "y" del x) = 0 + 6x(2y) = 12xy    ....(2)

From (1) and (2)

(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)

Concept: Applications of Partial Derivatives
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#### APPEARS IN

Chapter 6 Applications of Differentiation
Miscellaneous Problems | Q 9 | Page 156
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