IGNOU PHE 5 2026 - Determine all the first and second order partial derivatives for the f

Question

Determine all the first and second order partial derivatives for the function:

$$u(x, y) = x^2 \sin y + y^2 \cos x$

29 Jan 2026

Answer :

Word Count : 380

Let ( f(x,y) ) be a function of two independent variables ( x ) and ( y ). The first order partial derivatives measure the rate of change of the function with respect to one variable while keeping the other variable constant. The first order partial derivative of ( f ) with respect to ( x ) is defined as [ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x,y)}{h} ] where ( y ) is treated as a constant. Similarly, the first order partial derivative with respect to ( y ) is [ \frac{\partial f}{\partial y} = \lim_{k \to 0} \frac{f(x, y+k) - f(x,y)}{k} ] where ( x ) is treated as a constant. The second order partial derivatives are obtained by differentiating the first order

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