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Question Number 2111 by 123456 last updated on 03/Nov/15
lets f:R^2 →R  does  (∂f/∂x)=(∂/∂x)∫_0 ^x (∂f/∂y)dy   ?
$$\mathrm{lets}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R} \\ $$$$\mathrm{does} \\ $$$$\frac{\partial{f}}{\partial{x}}=\frac{\partial}{\partial{x}}\underset{\mathrm{0}} {\overset{{x}} {\int}}\frac{\partial{f}}{\partial{y}}{dy}\:\:\:? \\ $$
Answered by prakash jain last updated on 03/Nov/15
f=x^2 y  LHS=2xy  RHS=2x  LHS≠RHS  LHS will be dependent on y.  RHS will  NOT be dependent on y.
$${f}={x}^{\mathrm{2}} {y} \\ $$$$\mathrm{LHS}=\mathrm{2}{xy} \\ $$$$\mathrm{RHS}=\mathrm{2}{x} \\ $$$$\mathrm{LHS}\neq\mathrm{RHS} \\ $$$$\mathrm{LHS}\:\mathrm{will}\:\mathrm{be}\:\mathrm{dependent}\:\mathrm{on}\:{y}. \\ $$$$\mathrm{RHS}\:\mathrm{will}\:\:\mathrm{NOT}\:\mathrm{be}\:\mathrm{dependent}\:\mathrm{on}\:{y}. \\ $$

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